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NEET Physics Nuclei Notes

Nuclei

Nucleons. Protons and neutrons which are present in the nuclei of the atoms are collectively known as nucleons.

Atomic number. The number of protons present in the nucleus is called the atomic number.

It is denoted by Z.

Mass number. The total number of protons and neutrons present in the nucleus is called the mass number of the element. It is denoted by A.

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Number of protons in an atom = Z, Number of electrons in an atom = Z

Number of nucleons in an atom = A, Number of neutrons in an atom = N = A – Z.

Nuclear mass. The total mass of the protons and neutrons present in the nucleus is called the nuclear mass.

Isotopes. Nuclei having the same atomic number but different mass numbers are called isotopes.

Example: Hydrogen has three isotopes

\({ }_1^1 H,{ }_1^2 H \text { and }{ }_1^3 H \text {. }\)

Isobars. Nuclei having different atomic numbers but the same mass number are called isobars.

They contain different numbers of protons hence they are atoms of different elements.

\(\text { Example: (1) }{ }_1^3 \mathrm{H} \text { and }{ }_2^3 \mathrm{He}(2){ }_{18}^{40} \mathrm{Ar} \text { and }{ }_{20}^{40} \mathrm{Ca}\)

Isotones. Nuclei having an equal number of neutrons but different atomic numbers are called isotones. \({ }_1^3 \mathrm{H} \text { and }{ }_2^4 \mathrm{He}\) are isotones since they contain 2 neutrons.

Isomers. The nuclei having the same atomic number and same mass number but differ from one another in their internal structure and their nuclear energy states are called isomers. For example, the stable nucleus of \({ }_{38}^{87} \mathrm{Sr}\) has an isomer which emits gamma rays with a half life of 2.8 hour.

Mirror nuclei. Nuclei having the same mass number but the proton number and neutron number interchanged are called mirror nuclei. For example, \({ }_1^3 \mathrm{H} \text { and }{ }_2^3 \mathrm{He}\) are mirror nuclei.

Atomic mass unit. Atomic mass and nuclear mass are measured in terms of atomic mass unit (u). One atomic mass unit (1u) is defined as \(\frac{1}{12}^{t h}\) the mass of an atom of carbon- 12. Thus,

\(\begin{aligned}
& 1 \mathrm{u}=\frac{\text { mass of one atom of }{ }^{12} \mathrm{C}}{12} \\
& ∴ 1 \mathrm{u}=1.66 \times 10^{-27} \mathrm{~kg} \\
& ∴ \text { Mass of carbon-12 is, } 12 \mathrm{u} .
\end{aligned}\)

 

\(Mass of proton is, m_p=1.00727 u=1.00727 \times 1.66 \times 10^{-27}\) \(=1.67262 \times 10^{-17} \mathrm{~kg}\) \(Mass of neutron is, \mathrm{m}_n=1.00866 \mathrm{u}=1.6749 \times 10^{-27} \mathrm{~kg}\) \(Mass of electron is, \mathrm{m}_{\boldsymbol{c}}=0.00055 \mathrm{u}=9.13 \times 10^{-31} \mathrm{~kg}\)

Size of the nucleus. From experimental observations, it has been found that the volume of the nucleus is proportional to the number of nucleons present in it (or mass number A). Generally, nuclei are found to have a spherical shape.

\(R=R_0 A^{1 / 3}\)

Where Ro is a constant of proportionality. R = Ro when A = 1. Thus Ro represents the radius of the nucleus of a hydrogen atom which is nothing but a proton. From experiments, it has been found that Ro \(1.3 \times 10^{-15} \mathrm{~m}=1.3 \text { fermi }\)

Nuclear charge. If Z is the number of protons present in the nucleus then a charge of the nucleus is given by,

\(\mathrm{q}=+Z e, \mathrm{e}=1.602 \times 10^{-19} \mathrm{C}\)

Nuclear density. \(\text { Nuclear density }=\frac{\text { mass }}{\text { volume }}\)

Consider a nucleus of mass number A and let the mass of each nucleon is \(\mathrm{m}_{\mathrm{N}}=1.67 \times 10^{-27} \mathrm{~kg}\)

\(\text { Nuclear density }=\frac{A m_N}{\frac{4}{3} \partial R^3}=\frac{A m_N}{\frac{4}{3} \partial\left(R_0 A^{1 / 3}\right)^3}=\frac{A m_N}{\frac{4}{3} \partial R_0^3 A}\) \(=\frac{3 m_N}{4 \delta R_0^3}=\frac{3 \times 1.67 \times 10^{27}}{4 \delta \times\left(1.3 \times 10^{-15}\right)^3}=1.815 \times 10^{17} \mathrm{kgm}^{-3}\)

The nuclear density does not depend on mass number A or the number of nucleons in the nucleus. Hence nuclei of all elements have nearly the same density.

Mass energy relation. Before the special theory of relativity, it was presumed that mass and energy were conserved separately in a reaction. However, Einstein showed that mass is another form of energy and one can convert mass energy into other forms of energy.

Einstein’s mass-energy equivalence relation is given by, E=mc²

Here the energy equivalent of mass m is related by the above equation and c is the velocity of light in vacuum and is approximately equal to \(3 \times 10^8 \mathrm{~m} / \mathrm{s}\)

Energy Equivalent of One Atomic Mass Unit

\(∴\mathrm{lu}=931 \mathrm{MeV}\)

Mass defect. The difference between the sum of the masses of the constituent nucleons and the actual mass of the nucleus is called mass defect.

Let M be the mass of the nucleus having mass number A and atomic number Z. If mp is the mass of the proton and mn is the mass of the neutron, the mass defect of the nucleus is,

\(\Delta \mathrm{M}=\left[\mathrm{Zm}_{\mathrm{p}}+(\mathrm{A}-\mathrm{Z}) \mathrm{m}_{\mathrm{n}}\right]-\mathrm{M}\)

Binding energy. The binding energy of a nucleus can be defined as the minimum energy required to split the nucleus into its constituent nucleons.

Note. Mass defect is equivalent to binding energy.

\(\mathrm{E}_{\mathrm{b}}=\left[\mathrm{Z} \mathrm{m}_{\mathrm{p}}+(\mathrm{A}-\mathrm{Z}) \mathrm{m}_{\mathrm{n}}-\mathrm{M}\right] \mathrm{c}^2\)

The ratio of the binding energy of a nucleus to the number of nucleons in that nucleus is called binding energy per nucleon or specific binding energy.

\(\text { i.e., } E_{b a}=\frac{E_b}{A}\)

The binding energy per nucleon is close to the maximum value for nuclei in the medium mass number range of 30 to 170. Hence they have great stability. Binding energy per nucleon is maximum for \({ }_{26}^{56} \mathrm{Fe}\)

For higher mass numbers, the specific binding energy is lower and hence they are less stable.

For example, the last few naturally available elements exhibit radioactivity because of lesser stability.

The nature of the binding energy curve gives a clue to the release of energy in a nuclear process.

For example, lighter nuclei having lesser specific binding energy undergo fusion to form a nucleus of higher specific binding energy, resulting in the release of energy. Very heavy nuclei undergo fission to form medium-sized nuclei having greater specific binding energy, resulting in the release of energy.

Packing fraction: The mass defect per nucleon is called the packing fraction.

\(f=\frac{\Delta m}{A}=\frac{M-A}{A}\)

M = mass of the nucleus

A = mass number

The packing fraction measures the stability of nucleons. Smaller the value of packing fraction, larger is the stability of the nucleons.

Nuclear force. We know that for average mass nuclei the binding energy per nucleon is approximately 8MeV, which is much larger than the binding energy in atoms. Therefore to bind a nucleus together there must be a strong attractive force of a totally different kind. It must be strong enough to overcome the repulsion between the protons to bind both protons and neutrons into a tiny nuclear volume.

Characteristics of Nuclear Force

• Nuclear forces are the strongest known forces in nature.
• Nuclear forces are short-range forces.
• Nuclear forces are charge-independent.
• Nuclear forces have the property of saturation.
• Nuclear forces are spin-dependent.
• Nuclear forces are exchange forces.
• Nuclear forces have a repulsive core.
• Nuclear forces are non-central.

Radioactivity. The phenomenon of spontaneous disintegration of the nuclei of heavy elements with the emission of certain radiation is called radioactivity.

Law of radioactive decay: The rate of disintegration of a radioactive substance at any instant of time is directly proportional to the number of atoms of that substance present at that instant of time.

\(
N=N_0 e^{-\lambda t}
when \mathrm{t}=\frac{1}{\lambda}, \quad \mathrm{N}=\mathrm{N}_0 e^{-1} \Rightarrow \mathrm{N}=\frac{1}{e} \mathrm{~N}_0=0.3679 N_0\)

Therefore decay constant of a radioactive substance is defined as the reciprocal of the time during which the number of atoms of the substance decreases times the number of atoms originally present.

Half-life. Half life of a radioactive substance is defined as the time during which half of the original atoms disintegrate.

∴\(\mathrm{T}_{12}=\frac{0.693}{\lambda}\)

Mean life. The mean life or average life of a radioactive substance is the ratio of the sum of lives of all the individual atoms to the total number of atoms present in the sample \((\tau)\).

\(\tau=\frac{1}{\lambda}\)

Activity. The activity of a radioactive sample is a measure of the number of disintegrations per second or the rate of disintegrations in it.

Since the magnitude of the rate of disintegration is, \(\left|\frac{d N}{d t}\right|=\lambda N\) , i.e., we have activity, A= \(\mathrm{A}=\lambda N\)

The S.I. unit of activity is becquerel (Bq).

Activity is one becquerel if there is one disintegration per second in the substance.

The commonly used unit is curie (Ci).

One curie is the activity of a radioactive sample in which atoms disintegrate per second. It is also the activity of one gram of radium.

∴\(1 \mathrm{Ci}=3.7 \times 10^{10} \text { disintegrations } / \text { second }=3.7 \times 10^{10} \mathrm{~Bq} \text {. }\)

Note:

1. Radioactivity is a nuclear phenomenon. It is not affected by external factors such as temperature, pressure, electric and magnetic fields, chemical reactions, etc. The activity depends only on the radioactive substance and the number of atoms taken.

2. If A0 is the initial activity and A is the activity of a substance at an instant of time t, then

\(\mathrm{A}=\mathrm{A}_0 \mathrm{e}^{\mathrm{it}} \text { and } \mathrm{t}=2.303 \log _{10}\left(\frac{\mathrm{A}_0}{\mathrm{~A}}\right)\)

Alpha decay. Alpha decay is a process in which a nucleus decays spontaneously emitting an alpha particle. These α-particles have discrete values of energy. When a nucleus decays with the emission of α-particle (helium nucleus), the product nucleus (daughter nucleus) has atomic number two less and mass number four less than that of the decaying nucleus (parent nucleus).

In general,

\(\begin{aligned}
& { }_z^A \mathrm{X} \rightarrow{ }_{\mathrm{z}-2}^{\mathrm{A}} \mathrm{Y}+{ }_2^4 \mathrm{He} \\
& \mathrm{Eg}:{ }_{92}^{238} \mathrm{U} \rightarrow{ }_{90}^{234} \mathrm{Th}+{ }_2^4 \mathrm{He}
\end{aligned}\)

This spontaneous decay is possible only when the total mass of decay products is less than the mass of initial nucleus. The decrease in mass during the decay appears as K.E. of the products.

The disintegration energy or Q-value of the reaction is the difference between the initial mass energy and the total mass-energy of the decay products.

\(\text { i.e., } Q=\left(M_X-M_Y-m_\alpha\right) c^2\)

Where MX, MY & mα are the masses of the initial nucleus, product nucleus, and alpha particle. Clearly Q>0 for exothermic processes such as α-decay.

Β Decay

β –decay is a process in which a nucleus decays spontaneously emitting an electron or positron. During the β –decay process of a nucleus, the mass number of the product nucleus remains the same but its atomic number changes by one.

A common example of β decay is

\(
{ }_{15}^{32} P \rightarrow{ }_{16}^{32} S+e^{-}+\bar{v}
and that of \beta^* decay is
{ }_{11}^{22} \mathrm{Na} \rightarrow{ }_{10}^{22} \mathrm{Ne}+e^{+}+v
\)

Nuclear fission. Nuclear fission is the process in which the nucleus of an atom of a heavy element breaks up into two nuclei of comparable masses with the release of a large amount of energy.

The most important neutron-induced nuclear reaction is fission. An example of fission is when a uranium isotope \({ }_{92}^{235} \mathrm{U}\) bombarded with a neutron breaks into two intermediate-mass nuclear fragments

\({ }_0^1 n+{ }_{92}^{235} U \rightarrow{ }_{92}^{236} U \rightarrow{ }_{56}^{144} \mathrm{Ba}+{ }_{36}^{89} \mathrm{Kr}+3{ }_0^1 n\)

The same reaction can produce other pairs of intermediate mass fragments

\(\begin{gathered}
{ }_0^1 n+{ }_{92}^{235} \mathrm{U} \rightarrow{ }_{92}^{236} \mathrm{U} \rightarrow{ }_{51}^{133} \mathrm{Sb}+{ }_{41}^{99} \mathrm{Nb}+4{ }_0^1 n \text { or, } \\
{ }_0^1 n+{ }_{92}^{235} \mathrm{U} \rightarrow{ }_{92}^{236} \mathrm{U} \rightarrow{ }_{54}^{140} \mathrm{Xe}+{ }_{38}^{94} \mathrm{Sr}+2{ }_0^1 n
\end{gathered}\)

1. Controlled chain reaction. In this type, the chain reaction is first accelerated so that the neutron population is built up to a certain level and thereafter the number of fission-producing neutrons is kept constant. As a result, the energy is released in a controlled manner at a constant rate. This forms the principle of a nuclear reactor.

2. Uncontrolled chain reaction. In this type, the number of neutrons is allowed to multiply indefinitely by increasing the frequency of fissions so that the entire energy is released in a very short interval of time. The energy released will be uncontrolled and it results in a violent explosion. This forms the principle of an atom bomb.

An average of 2.5 neutrons are released in a single fission of . In order to achieve a self-propagating nuclear reaction, at least one of the neutrons in the fission must be captured by another \(\) nucleus and cause further fission. This possibility is determined by a quantity called reproduction constant or multiplication factor denoted by K. It is defined as the ratio of the
secondary neutrons produced to the original neutrons.

\(\text { i.e., } K=\frac{\text { number of neutrons in one event }}{\text { number of neutrons in the preceding event }}\)

Nuclear reactor. A nuclear reactor is a device in which a controlled nuclear chain reaction can be initiated and sustained to harness nuclear energy for constructive purposes.

The essential components of a nuclear reactor are:

Nuclear fuel. The core of the reactor is the site of nuclear fission. It contains the fuel elements in suitably fabricated form. The fuel may be \({ }_{94}^{239} \mathrm{Pu}\) or enriched uranium (i.e., one that has a greater abundance of \({ }_{92}^{235} U\) than naturally occurring uranium).

1. Moderator. The average energy of a neutron produced in fission of \({ }_{92}^{235} U\) is 2MeV. These neutrons unless slowed down will escape from the reactor without interacting with the uranium nuclei, unless a very large amount of fissionable material is used for sustaining the chain reaction. What one needs to do is to slow down the fast neutrons by elastic scattering with light nuclei. Therefore, in reactors, light nuclei called moderators are provided along with the fissionable nuclei for slowing down fast neutrons. The moderators commonly used are water, heavy water (D2O) and graphite.

2. Control rods. The reaction can be shut down by means of control rods (made of, for example, cadmium) that have high absorption of neutrons. The K value can be varied in a reactor by the proper use of these rods, hence are called control rods.

3. Cooling system. The energy (heat) released in fission is continuously removed by a suitable coolant. The coolant transfers heat to a working fluid which in turn may produce steam. The steam drives turbines and generates electricity. The coolants used are carbon dioxide gas or ordinary water when graphite is the moderator. Heavy water is used also as coolant when it is used as moderator.

4. Reflector. The core is surrounded by a reflector to reduce leakage. This helps in the reduction of critical size. The commonly used reflector is graphite.

5. Safety system & protective shield. The problems of reactor safety are very vast and complex. All reactors are provided with back up cooling system as a safety measure. This system takes over if the regular cooling system fails.

A reactor is provided with adequate shielding to minimize the biological effects of harmful radiation ( -rays and neutrons ). The shield is generally a 2m thick concrete wall surrounding the reactor.

Nuclear fusion. Nuclear fusion is the opposite of nuclear fission. It is the process in which two lighter nuclei fuse together to form a heavier nucleus with the liberation of energy

\({ }_1^1 \mathrm{H}+{ }_1^1 \mathrm{H} \rightarrow{ }_1^2 \mathrm{H}+\mathrm{e}^{+}+\mathrm{v}+0.42 \mathrm{MeV}\)

Controlled thermonuclear fusion. In controlled fusion reactors, the aim is to generate steady power by heating the nuclear fuel to a temperature in the range of 108 K. At these temperatures, the fuel is a mixture of positive ions and electrons (plasma). The challenge is to combine this plasma, since no container can stand such a high temperature. This can be done by using a magnetic bottle.

NEET Physics Atoms Notes

Atoms

Important facts about Rutherford’s α–α-scattering experiment:

• Most of the α–particles do not suffer collisions with the gold foil.
• Only about 0.14% of the incident α–particles scatter by more than 10.
• About 1 in 8000 α–particles deflect by more than 900.

Graph of the number of alpha particles scattered versus scattering angle

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Rutherford argued that, for large angle deflection there must be a massive positively charged body in every atom where the total positive charges of the atom are concentrated.

Hence Rutherford is credited with the discovery of the nucleus.

Impact Parameter

It is the perpendicular distance of the initial velocity vector of the α–particle from the center of the nucleus.

The impact parameter is given by,

\(\mathrm{b}=\frac{\mathrm{Ze}^2 \cot \left(\frac{\theta}{2}\right)}{4 \pi \mathrm{E}_0\left(\frac{1}{2} m v^2\right)} \Rightarrow \mathrm{b} \propto \cot \left(\frac{\theta}{2}\right)\)

The trajectory of α–particle depends on the impact parameter.

In case of a head-on collision, the impact parameter is minimum and the α–particle rebounds back.

For a large impact parameter, the α–particle goes nearly undefeated.

Distance of closest approach: At the distance of closest approach entire K.E. of the alpha particle gets converted into electrostatic P.E.

i,e., \(\begin{aligned}
\frac{1}{2} \mathrm{mv}^2 & =\left(\frac{1}{4 \pi \varepsilon_0}\right) \frac{(\mathrm{Ze})(2 \mathrm{e})}{\mathrm{r}_0} \\
\mathrm{r}_0 & =\frac{9 \times 10^9 \times 4 \mathrm{Ze}^2}{m v^2}
\end{aligned}\)

Bohr’s Quantum Condition

The angular momentum of an electron in the nth stationary orbit is given by,

\(\mathrm{mvr}=\mathrm{n} \frac{\mathrm{h}}{2 \pi} \quad \mathrm{n}=1,2,3, \ldots \ldots \ldots\)

When an electron makes a transition from a higher energy state to a lower energy state, the difference in energy is emitted in the form of electromagnetic radiation.

\(\mathrm{E}_2-\mathrm{E}_1=\mathrm{hv}\)

The orbital radius of electrons in ‘n^th’ orbit is given by:

\(\begin{aligned}
& \mathrm{r}_{\mathrm{n}}=\frac{\mathrm{n}^2 \mathrm{~h}^2 \varepsilon_0}{\pi \mathrm{mZ} \mathrm{e}^2} \\
& \mathrm{r}_{\mathrm{n}} \propto \frac{\mathrm{n}^2}{\mathrm{Z}}
\end{aligned}\)

For hydrogen atom, \(r_n \propto n^2\)

The radius of n^th orbit in a hydrogen atom is,

\(\mathrm{r}_{\mathrm{n}}=0.53 \mathrm{n}^2\) A

The orbital velocity of electrons is given by:

\(\begin{aligned}
& v_{\mathrm{n}}=\frac{\mathrm{Ze}^2}{2 \varepsilon_0 \mathrm{nh}} \\
& \mathrm{v}_{\mathrm{n}} \propto \frac{\mathrm{Z}}{\mathrm{n}} \\
& \mathrm{v}_{\mathrm{n}}=\frac{\mathrm{Z}}{\mathrm{n}} \frac{\mathrm{c}}{137}
\end{aligned}\)

Where ‘c’ is the speed of light.

In the case of the hydrogen atom,

\(\mathrm{v}_{\mathrm{n}}=\frac{1}{\mathrm{n}} \frac{\mathrm{c}}{137}\) \(\text { If } \mathrm{n}=1, \mathrm{v}_{\mathrm{n}}=\frac{\mathrm{c}}{137} \simeq 2.2 \times 10^6 \mathrm{~ms}^{-1}\)

Expression for total energy

\(\begin{aligned}
& E_n=-\frac{1}{n^2} \frac{m Z^2 e^4}{8 \varepsilon_0^2 h^2} \\
& E_n \propto \frac{Z^2}{n^2} \\
& E_n=(-13.6) \frac{Z^2}{n^2} e V
\end{aligned}\)

For hydrogen atoms,

\(\mathrm{E}_{\mathrm{n}}=\frac{-13.6}{\mathrm{n}^2} \mathrm{eV}\)

If n = 1, then,

E = –13.6 eV

∴ The ionization energy of the hydrogen atom in the ground state is 13.6 eV & its ionization potential is 13.6 V.

\(\begin{aligned}
& \text { K.E. }=+\frac{1}{\mathrm{n}^2} \frac{\mathrm{mZ^{2 }} \mathrm{e}^4}{8 \varepsilon_0^2 \mathrm{~h}^2} \\
& \text { P.E. }=-\frac{1}{\mathrm{n}^2} \frac{\mathrm{mZ}^2 \mathrm{e}^4}{4 \varepsilon_0^2 \mathrm{~h}^2} \\
& |\mathrm{E}|=\text { K.E.; P.E. }=2 \mathrm{E}
\end{aligned}\) \(\text { wave number, } \quad \frac{1}{\lambda}=R Z^2\left[\frac{1}{\mathrm{n}_1^2}-\frac{1}{\mathrm{n}_2^2}\right]\)

Energy Level Diagram For Hydrogen

No. of spectral lines emitted is given by, \(\frac{\mathrm{n}(\mathrm{n}-1)}{2}\)

The time period of revolution of an electron is given by,

\(\begin{aligned}
& T=\frac{2 \pi r}{v} \\
& T \propto \frac{r}{v} \Rightarrow T \propto \frac{n^2}{\frac{1}{n}} \Rightarrow T \propto n^3 \\
& f \propto \frac{1}{T} \Rightarrow f \propto \frac{v}{r} \Rightarrow f \propto \frac{1}{n^3}
\end{aligned}\)

The wavelength of light emitted by hydrogen atoms is,

\(\lambda=\frac{12,420}{\Delta \mathrm{E}(\mathrm{eV})}^{\circ} \mathrm{A}\)

The atom stays in the excited state for about 10 nanoseconds.

Rydberg’s constant is not the same for all elements.

It is the same for elements having the same number of electrons.

The permitted value of ‘n’ ranges up to. However, only values up to 7 have so far been observed.

Balmer series was the first observed spectral series.

The B.E. of the electron in the ground state of hydrogen is called rydberg.

⇒ 1 rydberg = 13.6 eV

NEET Physics Dual Nature of Radiation and Matter Notes

Dual Nature of Radiation and Matter

The energy gained by an electron when it is accelerated between a potential difference of 1 volt is
called 1eV.

The minimum energy required by an electron to escape from the metal surface is called work
function \(\left(\phi_0\right)\)

Caesium (Cs) has the least work function (2.14 eV)

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Platinum (Pt) has the highest work function (5.65 eV)

The phenomenon of emission of free electrons from a metal surface on illumination of electromagnetic radiation of suitable frequency is called Photoelectric Effect.

Maximum kinetic energy of emitted photoelectrons is given by,

\(K_{\max }=e V_0\)

where, V₀ is stopping potential.

Experimental Observations of Photoelectric Effect

1. Above threshold frequency, the photocurrent is directly proportional to intensity of incident light.

2. Saturation current is proportional to intensity of incident radiation but stopping potential is independent of intensity.

3. There exists a certain minimum frequency below which photoemission is absent is called threshold frequency.

4. Photoelectric effect is an instantaneous process.

Note: Wave theory of light failed to explain experimental observations of photoelectric effect.

Einstein’s Explanation for Photoelectric Effect

According to Einstein when a photon of energy falls on a metal surface, the maximum kinetic energy of emitted photoelectrons is given by,

\(K_{\max }=h v-\phi_0 \rightarrow(1)\)

Where, is the work function of the metal.

Equation (1) is in the form of y = mx + c

i.e., the graph of \(K_{\max }\)versus will be a straight line with slope h.

\(\begin{aligned}
(1) \Rightarrow e V_0 & =h v-\phi_0 \quad\left(∵ K_{\max }=e V_0\right) \\
V_0 & =\left(\frac{h}{e}\right) v-\frac{\phi_0}{e} \rightarrow(2)
\end{aligned}\)

Equation (2) is also in the form of y = mx + c

i.e., the graph V0 of versus is a straight line with slope \(\frac{h}{e}\)

Important properties of photon

• Momentum of the photon is \(\frac{E}{c}=\frac{h v}{c}\)
• The rest mass of a photon is zero.
• Photons are electrically neutral; they are not deflected by electric or magnetic fields.

Matter waves

Waves associated with moving matter are called matter waves.

According to Louis de Broglie wavelength of matter waves is given by,

\(\lambda=\frac{h}{P}=\frac{h}{m v}\)

Consider a particle of mass m and charge q accelerated from rest through a potential V. Then
The kinetic energy of the charged particle is given by,

K = qV

\(\text { w.k.t., } \lambda=\frac{h}{P}=\frac{h}{\sqrt{2 m K}}\) \(\Rightarrow \lambda=\frac{h}{\sqrt{2 m q V}}\)

For an electron,

\(\lambda=\frac{h}{\sqrt{2 m e V}}=\frac{1.227}{\sqrt{V}} \mathrm{~nm}\)

Note: Davisson and Germer’s experiment proved the wave nature of electrons.

According to Heisenberg’s uncertainty principle, it is impossible to measure two canonically conjugate physical quantities like position and momentum of a microscopic entity simultaneously and accurately.

Or in simple form,

It is impossible to measure position and momentum of a microscopic particle accurately and simultaneously.

Electromagnetic Waves

Maxwell modified the Ampere’s circuital law \(\oint \overrightarrow{\mathrm{B}} \cdot \mathrm{d} \vec{\ell}=\mu_0 \mathrm{I}\) to explain the continuity of current in a circuit containing a capacitor.

⇒ \(\phi=\mathrm{EA}=\frac{\sigma}{\mathrm{s}_0} \mathrm{~A}=\frac{\mathrm{Q}}{\mathrm{A} \varepsilon_0} \mathrm{~A}\)

⇒ \(\frac{\mathrm{d} \phi}{\mathrm{dt}}=\frac{1}{\varepsilon_0} \frac{\mathrm{dQ}}{\mathrm{dt}}\)

⇒ \(\mathrm{I}_{\mathrm{d}}=\frac{\mathrm{dQ}}{\mathrm{dt}}=\varepsilon_0 \frac{\mathrm{dQ}}{\mathrm{dt}}\)

 

This was the missing term suggested by Maxwell called displacement current Id.

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Displacement current is due to the rate of change of electric field between the plates of the capacitor

Conduction current and the displacement in a circuit may not be continuous but their sum is always continuous.

A modified form of Ampere’s circuital law / Ampere–Maxwell law is,

⇒ \(\oint \overrightarrow{\mathrm{B}} \cdot \mathrm{d} \vec{\ell}-\mu_0 \mathrm{I}+\varepsilon \mu_0 \frac{\mathrm{d} \phi_1}{\mathrm{dt}}\)

 

Maxwell’s Equations

 

\(1. \oint \vec{E} \cdot d \vec{A}=\frac{Q}{e_0} (Gauss’ law of electricity)\)

 

\(2. \oint \overrightarrow{\mathrm{B}} \cdot \mathrm{d} \overrightarrow{\mathrm{A}}=0 (Gauss’ law for magnetism)\)

 

\(3. \oint \overrightarrow{\mathrm{E}} \cdot \mathrm{d} \vec{\ell}=-\frac{\mathrm{d} \phi_{\mathrm{B}}}{\mathrm{dt}} (Faraday’s law)\)

 

\(4. \oint \overrightarrow{\mathrm{B}} \cdot \mathrm{d} \vec{\ell}-\mu_0 \mathrm{i}_{\mathrm{c}}+\mu_0 \varepsilon_0 \frac{\mathrm{d} \phi}{\mathrm{dt}} (Ampere – Maxwell law)\)

 

Representation of An Electromagnetic Wave:

\(\begin{aligned}
& E_{\mathrm{z}}=\mathrm{E}_0 \sin (\mathrm{kz}-\omega \mathrm{t}) \\
& \mathrm{B}_{\mathrm{y}}=\mathrm{B}_0 \sin (\mathrm{kz}-\omega \mathrm{t})
\end{aligned}\)

The direction of the EM wave is given by \(\overrightarrow{\mathrm{E}} \times \overrightarrow{\mathrm{B}}\)

\(\begin{aligned}
\frac{\mathrm{E}_0}{\mathrm{~B}_0} & =\mathrm{c} \\
\mathrm{k} & =\frac{2 \pi}{\lambda}, \omega=2 \pi \mathrm{f} \\
\frac{\omega}{\mathrm{k}} & =\frac{2 \pi \mathrm{f}}{2 \pi} \lambda=\mathrm{c} \\
\mathrm{c} & =\frac{1}{\sqrt{\mu_0 \varepsilon_0}}
\end{aligned}\)

The velocity of light in a medium is given by,

\(\begin{aligned}
& \mathrm{v}=\frac{1}{\sqrt{\mu \varepsilon}}=\frac{1}{\sqrt{\mu_0 \mu_{\mathrm{r}} \varepsilon_0 \varepsilon_{\mathrm{r}}}} \\
& \mathrm{v}=\frac{1}{\sqrt{\mu_0 \varepsilon_0}} \times \frac{1}{\sqrt{\mu_{\mathrm{r}} \varepsilon_{\mathrm{t}}}} \\
& \mathrm{v}=\frac{\mathrm{c}}{\sqrt{\mu_z \varepsilon_{\mathrm{r}}}}
\end{aligned}\)

The rate of energy transported per unit area by EM wave is given by,

Pointing vector, \(\overrightarrow{\mathrm{S}}=\frac{\mathrm{EB}}{\mu_0}=\frac{\mathrm{E}^2}{\mu_0 \mathrm{c}}\)

Energy density of the electric field is given by,

\(u_E=\frac{1}{2} \varepsilon_0 E^2\)

Energy density of the magnetic field is given by,

\(\mathrm{u}_{\mathrm{B}}=\frac{\mathrm{B}^2}{2 \mu_0}\)

• Average electric energy density = average magnetic energy density.
• The intensity of the EM wave is given by,

\(\mathrm{I}=\mathrm{U}_{\mathrm{m}} \mathrm{c}=\frac{\mathrm{B}_0^2}{2 \mu_0} \mathrm{c}=\frac{1}{2} \varepsilon_0 \mathrm{E}_0^2 \mathrm{c}\)

• The electric vector is responsible for all optical effects. This vector is also known as light vector.
• The pressure exerted by an EM wave is given by,

\(\mathrm{P}=\frac{\mathrm{I}}{\mathrm{c}}\)

• The momentum carried by an EM wave is given by,

\(\mathrm{P}=\frac{\mathrm{U}}{\mathrm{c}}\)

NEET Physics Electrostatic Potential and Capacitance Notes

Electrostatic Potential and Capacitance

The electric potential at a point is defined as the work done in bringing a unit positive charge (with uniform speed) from infinity to that point against the electrostatic force of the field.

V = \(\frac{W}{q}\)

Charges always flow from a body at a higher potential to a body at a lower potential.

Electric potential due to a point charge is given by,

\(\mathrm{V}=\frac{1}{4 \pi \varepsilon_0} \frac{\mathrm{q}}{\mathrm{r}}\)

Read And Learn More: NEET Physics Notes

Electric potential is a scalar quantity.

\(1 \text { volt }=\frac{1 \text { joule }}{1 \text { coulomb }} \Rightarrow 1 \mathrm{~V}=\frac{1 \mathrm{~J}}{1 \mathrm{C}}\)

The electric potential at any general point due to a dipole is given by

\(\mathrm{V}=\frac{1}{4 \pi \varepsilon_0} \frac{\mathrm{p} \cos \theta}{\mathrm{r}^2}\)

Where ‘p’ is electric dipole moment θ, is the angle between \(\overrightarrow{\mathrm{r}} \text { and } \overrightarrow{\mathrm{p}}\).

A surface with the same value of potential at all points on the surface is known as an equipotential surface.

The p.d. between any two points on the equipotential surface is zero.

The work done in transferring a charge from one point to another on an equipotential surface is zero.

Electric Field and Potential Are Related as

E = \(\frac{dV}{dr}\)

Where \(\frac{dV}{dr}\) is known as a potential gradient.

Potential energy of a system of two charges is given by,

\(\mathrm{U}=\frac{1}{4 \pi \varepsilon_0} \frac{\mathrm{q}_1 \mathrm{q}_2}{\mathrm{r}_{12}}\)

Where r₁₂is the distance between two charges.

Potential energy of a system of three charges is given by,

\(\mathrm{U}=\frac{1}{4 \pi \varepsilon_0}\left[\frac{\mathrm{q}_1 \mathrm{q}_2}{\mathrm{r}_{12}}+\frac{\mathrm{q}_1 \mathrm{q}_3}{\mathrm{r}_{13}}+\frac{\mathrm{q}_2 \mathrm{q}_3}{\mathrm{r}_{23}}\right]\)

In general, for ‘n’ charges.

\(\mathrm{U}=\frac{1}{4 \pi \varepsilon_0} \sum_{\text {alpuis }} \frac{\mathrm{q}_{\mathrm{i}} \mathrm{q}_{\mathrm{j}}}{\mathrm{r}_{\mathrm{ij}}}\)

The atoms or molecules in which the effective positive charge center and negative charge center do not coincide are called polar molecules.

A dielectric made of polar molecules is called a polar dielectric. The atoms or molecules in which the effective positive charge center and negative charge center coincide are called non–polar molecules.

A dielectric made of non–polar molecules is called non–polar dielectric.

The dipole moment per unit volume of the dielectric is called polarisation.

Polarisation, \(P=\chi_e E\)

Where X_eis known as the electrical susceptibility of the dielectric medium.

When a very high electric field is applied to the insulator, the force experienced by the valence electrons may be large enough to get pulled from the atom. Now electrons constitute current inside the insulator. This is known as the breakdown of dielectric or the breakdown of an insulating material.

The minimum value of an electric field that produces a breakdown of the dielectric is known as the dielectric strength of the dielectric.

Reason for Lightening

When the electric field developed by the charged cloud becomes greater than the dielectric strength of air, then there will be a huge electric discharge. This is the reason for lightening.

The dielectric strength of dry air is about 3×10⁶V/m.

The capacitor is a device that stores energy in the form of an electric field.

Or

It is a system of two conductors separated by a distance used to store electric charge.

The capacitance of a capacitor ‘C’ is given by

\(\frac{Q}{V}\)

Where ‘Q’ is the charge stored and ‘V’ is the potential difference.

Note:

A single conductor also has the capacity to store charges.

The capacitance of a spherical conductor is given by,

\(\mathrm{C}=\frac{\mathrm{Q}}{\mathrm{V}}=\frac{\mathrm{Q}}{\frac{1}{4 \pi \varepsilon_0} \frac{\mathrm{Q}}{\mathrm{R}}}\) \(\mathrm{C}=4 \pi \mathrm{x}_0 \mathrm{R}\)

Where ‘R’ is the radius of the conductor.

The capacitance of a parallel plate air capacitor is given by,

\(C=\frac{\varepsilon_p A}{d}\)

Where ‘A’ is the area of capacitor plates and ‘d’ is the distance between capacitor plates.

When a parallel plate capacitor is filled with a dielectric of dielectric constant ‘K’ then,

\(C=K\left(\frac{\varepsilon_0 A}{d}\right)\)

i.e., the capacitance increases K times.

The capacitance of a capacitor when it is partially filled with dielectric

\(C=\frac{\varepsilon_0 A}{(d-t)+\frac{t}{K}}\)

Where ‘t’ is the thickness of the dielectric slab.

The capacitance of a parallel plate capacitor with a conducting slab.

\(\mathrm{C}=\frac{\varepsilon_0 \mathrm{~A}}{\mathrm{~d}-\mathrm{t}}\)

(∵ for a conducting slab K= ∞)

Series Combination of Capacitors

When capacitors are connected in series charge stored in all the capacitors is the same.

The effective capacitance is,

\(\frac{1}{C_5}=\frac{1}{C_1}+\frac{1}{C_2}+\ldots . .+\frac{1}{C_E}\)

When two capacitors of capacitance C₁ and C₂ are connected in series, then,

\(\frac{1}{C_s}=\frac{1}{C_1}+\frac{1}{C_2}=\frac{C_1+C_2}{C_1 C_2}\) \(\mathrm{C}_{\mathrm{S}}=\frac{\mathrm{C}_1 \mathrm{C}_2}{\mathrm{C}_1+\mathrm{C}_2}\)

When ‘n’ capacitors of equal capacitance ‘C’ are connected in series, then

\(\mathrm{C}_{\mathrm{S}}=\frac{\mathrm{C}}{\mathrm{n}}\)

Parallel Combination of Capacitors

When ‘n’ capacitors are connected in parallel, the p.d. between each capacitor will be the same.

The effective capacitance is,

C_p= C₁ + C₂ + …. + C_n

When ‘n’ capacitors of equal capacitance ‘C’ are connected in parallel, then

\(\mathrm{C}_{\mathrm{p}}=\mathrm{nC}\)

The p.d. between two spherical shells of a spherical capacitor is given by,

\(V=\frac{Q}{4 \pi \varepsilon_0}\left[\frac{1}{a}-\frac{1}{b}\right]\)

Where Q is the magnitude of charge on either shell, a is the radius of the inner shell and b is the radius of the outer shell.

∴ The capacitance of the spherical capacitor is,

\(\begin{aligned}
& C=\frac{Q}{V} \\
& C=\frac{4 \pi \varepsilon_0 a b}{b-a}
\end{aligned}\)

Energy stored in a capacitor is given by:

\(\mathrm{U}=\frac{\mathrm{Q}^2}{2 \mathrm{C}}=\frac{1}{2} \mathrm{CV}^2=\frac{1}{2} \mathrm{QV}\)

Note:

When a battery supplies charge to a capacitor, only 50% of the work done by the battery gets stored as energy. The remaining 50% is dissipated in the form of heat.

The Energy Density in a Parallel Plate Capacitor

\(\)

u = \(\frac{U}{V}\)

\(\mathrm{u}=\frac{\frac{1}{2} \mathrm{CV}^2}{\mathrm{Ad}}=\frac{1}{2} \frac{\varepsilon_{\mathrm{g}} \mathrm{A} \mathrm{V}^2}{\mathrm{~d}(\mathrm{Ad})}\)

\(\mathrm{u}=\frac{1}{2} \varepsilon_0 \mathrm{E}^2\) (∴ E = \(\frac{V}{d}\)

Common potential when two capacitors are connected in parallel

\(\mathrm{V}_{\text {coev }}=\frac{\mathrm{C}_1 \mathrm{~V}_1+\mathrm{C}_2 \mathrm{~V}_2}{\mathrm{C}_1+\mathrm{C}_2}\)

The energy loss is given by,

\(\mathrm{U}_{\mathrm{leas}}=\frac{1}{2}\left(\frac{\mathrm{C}_1 \mathrm{C}_2}{\mathrm{C}_1+\mathrm{C}_2}\right)\left(\mathrm{V}_1-\mathrm{V}_2\right)^2\)

NEET Physics Ray Optics and Optical Instruments Notes

Ray Optics and Optical Instruments

Laws of Reflection of Light:

1. The incident ray, reflected ray and the normal drawn at the point of incidence all lie in the same plane.
2. Angle of incidence is equal to the angle of reflection.

\(\text { i.e., }\lfloor i=\lfloor r\)

Sign conventions:

• All distances are measured from the pole of the mirror.
• Distances measured in the direction of incident light are taken positive and vice versa.
• Heights measured upward and perpendicular to the principal axis are taken positive and vice versa.

Read And Learn More: NEET Physics Notes

For a spherical mirror, the relation between focal length and radius of curvature is,

\(\mathrm{f}=\frac{\mathrm{R}}{2}\)

Where, f is the focal length and ‘R’ is the radius of curvature of the mirror.

Mirror equation

\(\frac{1}{f}=\frac{1}{u}+\frac{1}{v}\)

Where u is the object distance and v is the image distance.

Linear magnification is the ratio of the height of the image to the height of the object.

\(\mathrm{m}=\frac{\mathrm{h}_{\mathrm{i}}}{\mathrm{h}_{\mathrm{o}}}\)

For a spherical mirror, \(\mathrm{m}=-\frac{\mathrm{v}}{\mathrm{u}}\)

The refractive index of a medium is defined as the ratio of the speed of light in vacuum (c) to the speed of light of medium (v)

\(\mathrm{n}=\frac{\mathrm{e}}{\mathrm{v}}\)

Refractive index of a medium depends on the wavelength of light used.

Longer the wavelength smaller is the refractive index.

Laws of refraction of light

1. The incident ray, refracted ray and the normal drawn at the point of incidence all lie in the same plane.
2. Sine of angle of incidence to the sine of angle of refraction is a constant for a given pair of media and for given wavelength of light.

\(\text { i.e., } \frac{\sin i}{\sin r}=\text { constant }=n_{21}\)

Where is the refractive index of medium 2 w.r.to medium 1.

If \(n_{21}>1, \mathrm{r}<\mathrm{i} \text { and if } n_{21}<1, \mathrm{r}>\mathrm{i}\)

Expression for normal shift

N. S .= \(t\left(1-\frac{1}{n}\right)\)

where t is the thickness and n is the refractive index of the medium.

Relation between refractive index and critical angle

\(\sin \mathrm{C}=\mathrm{n}_{21}=\frac{\mathrm{n}_2}{\mathrm{n}_1}\)

Where ‘C’ is the critical angle

Relation between n, u, v, and R for a spherical refracting surface.

\(\frac{\mathrm{n}_2}{\mathrm{v}}-\frac{\mathrm{n}_1}{\mathrm{u}}=\frac{\mathrm{n}_2-\mathrm{n}_1}{\mathrm{R}}\)

Where, n₂ is the refractive index of the image medium.

n₁ is the refractive index of the object medium.

And R is the radius of curvature.

Lens maker’s formula

\(\frac{1}{\mathrm{f}}=\left(\mathrm{n}_{21}-1\right)\left(\frac{1}{\mathrm{R}_1}-\frac{1}{\mathrm{R}_2}\right)\)

Thin lens formula

\(\frac{1}{v}-\frac{1}{u}=\frac{1}{f}\)

Magnification produced by a lens

\(\mathrm{m}=\frac{\mathrm{v}}{\mathrm{u}}\)

Power of a lens

\(P=\frac{1}{f}\)

The S.I. unit of power of a lens is dioptre (D)

1D = 1m¯¹

Combination of thin lenses in contact

In terms of power,

P_eff = P₁ + P₂ + P₃ + ….

In terms of magnification,

M_eff = m₁. m₂ . m₃…..

Refraction through a prism

At the angle of minimum deviation

\(\begin{aligned}
& \delta=\mathrm{D}_{\mathrm{m}}, \mathrm{i}=\mathrm{e} \Rightarrow r_{\mathrm{i}}=r_2 \\
& \Rightarrow \mathrm{r}=\frac{\mathrm{A}}{2} \\
& \mathrm{D}=\frac{\left(\mathrm{A}+\mathrm{D}_{\mathrm{m}}\right)}{2} \\
& \mathrm{n}_{21}=\frac{\mathrm{n}_2}{\mathrm{n}_1}=\frac{\sin \left(\frac{\left(\mathrm{A}+\mathrm{D}_{\mathrm{m}}\right)}{2}\right)}{\sin \left(\frac{\mathrm{A}}{2}\right)}
\end{aligned}\)

For a small-angle prism,

\(\begin{aligned}
& \mathrm{n}_{21}=\frac{\frac{\left(\mathrm{A}+\mathrm{D}_{\mathrm{m}}\right)}{2}}{\left(\frac{\mathrm{A}}{2}\right)} \\
& \& \mathrm{D}_{\mathrm{m}}=\left(n_{21}-1\right) \mathrm{A}
\end{aligned}\)

In the above set of equations i is the angle of incidence, e is the angle of emergence, A is angle of prism & are angle of refractions at two boundaries, is deviation and Dm is angle of minimum deviation.

According to Rayleigh’s law of scattering if the particle size is greater the then,

Scattering is proportional to \(\frac{1}{\lambda^4}\).

Standard value of near point of human eye is D = 25 cm.

Magnification produced by simple microscope when image is produced at near point,

When the image is formed at infinity,

\(m=\frac{D}{f}\)

Magnification produced by a compound microscope when image is formed at infinity is,

\(\begin{aligned}
& m=m_0 m_e \\
& m=\left(\frac{L}{f_0}\right)\left(\frac{D}{f_e}\right)
\end{aligned}\)

Magnification produced by a compound microscope when image is formed at near point is,

Where, and are the magnification of the objective and eyepiece respectively fo and fe are the focal lengths of the objective and the eyepiece respectively.

Magnification of the telescope is given by,

\(m=\frac{f_o}{f_o}\)

where fo and fe are the focal lengths of objective and eyepiece respectively.

Magnifying power of a telescope is the ratio of the angle subtended at the eye by the image to the angle subtended at the eye by the object

\(\text { i.e., } \mathrm{m}=\frac{\beta}{\alpha}\)

NEET Physics Alternating Current Notes

Alternating Current

Let the alternating emf is given by,

\(\varepsilon=\varepsilon_0 \sin \omega t\)

where, \(\varepsilon_0=\mathrm{NAB} \omega\)

The instantaneous value of current is given by

\(\mathrm{I}=\mathrm{I}_0 \sin \omega \mathrm{t}\)

Note:

\(I_m=\frac{I_0}{\sqrt{2}} ; v_{\max }=\frac{v_0}{\sqrt{2}}\)

or

\(\mathrm{I}_{\max }=0.707 \mathrm{I}_0 ; \mathrm{V}_{\max }=0.707 \mathrm{~V}_0\)

Where, I_rms & V_rms are effective values & I₀ & V₀ are the peak values.

\(\mathrm{I}_w=\frac{2}{\pi} \mathrm{I}_0=0.637 \mathrm{I}_0=(63.7 \%) \mathrm{I}_0\) \(\mathrm{v}_m=\frac{2}{\pi} \mathrm{V}_0=0.637 \mathrm{~V}_6=(63.7 \%) \mathrm{V}_0\)

Peak to peak value of AC is given by

2V₀ or 2I₀

Read And Learn More: NEET Physics Notes

Time Difference

If the phase difference between alternating current and voltage is then the time difference between them is,

\(\mathrm{TD}=\frac{\mathrm{T}}{2 \pi} \times \phi\)

The mean value of ac for the half cycle is,

\(I_{\text {mean }}=\frac{2}{\pi} I_0\)

The rms value of current is given by,

\(\mathrm{I}_{\max }=\frac{\mathrm{I}_0}{\sqrt{2}}\)

Alternating Voltage Applied to a Pure Resistor

\(\begin{aligned}
& \mathrm{V}=\mathrm{V}_0 \sin \omega \mathrm{t} \\
& \mathrm{I}=\mathrm{I}_0 \sin \omega t,
\end{aligned}\)

Where, \(I_0=\frac{V_0}{R}\)

V and I are in phase

Alternating Voltage Applied to a Pure Inductor

\(\begin{aligned}
& V=V_0 \sin \omega t \\
& I=I_0 \sin \left(\omega t-\frac{\pi}{2}\right)
\end{aligned}\)

Where, \(I_0=\frac{V_0}{\omega L}\)

The quantity is analogous to resistance, called inductive reactance.

Inductive reactance, \(\mathrm{X}_{\mathrm{L}}=\omega \mathrm{L}=2 \pi \mathrm{fL}\)

For dc, \(f=0, \Rightarrow X_L=0\)

i.e., the inductor behaves like a wire for dc.

The phase difference between V and I is —.

V leads I by \(\frac{\pi}{2}\)

Alternating Voltage Applied to a Pure Capacitor

\(\mathrm{V}=\mathrm{V}_0 \sin \omega \mathrm{t}
I=I_0 \sin \left(\omega t+\frac{\pi}{2}\right)
\)

Where, \(I_0=\frac{V_0}{\left(\frac{1}{\omega C}\right)}\)

The quantity \(\) is analogous to resistance called capacitive reactance.

Capacitive reactance, \(\left(\frac{1}{\omega \mathrm{C}}\right)\)

For dc, f = 0, X_c = ∞

The phase difference between V and I is \(\frac{\pi}{2}\).

V lags I by \(\frac{\pi}{2}\).

Alternating Voltage Applied to a Series LCR Circuit

The applied alternating voltage is,

\(\mathrm{V}=\mathrm{V}_0 \sin \omega \mathrm{t}\)

The current flowing in the circuit is,

\(\mathrm{I}=\mathrm{I}_0 \sin (\omega \mathrm{t}+\phi)\) \(\text { Where, } I_4=\frac{V_0}{\sqrt{R^2+\left(\frac{1}{\omega C}-\omega L\right)^2}}\) \(\text { And } \phi=\tan ^{-1}\left(\frac{X_C-X_L}{R}\right)\)

Expression for Average Power in AC Circuits

\(\overline{\mathrm{P}}=\mathrm{V}_{\mathrm{max}} \mathrm{I}_{\mathrm{max}} \cos \phi\)

The term \(\cos \phi\) is called power factor.

For pure resistive circuit, \(\phi=0 \text { i.e., } \cos \phi=1\)

For pure capacitive circuit, \(\phi=\frac{\pi}{2} \text { i.e., } \cos \phi=0\)

For pure inductive circuit, \(\phi=\frac{\pi}{2} \text { i.e., } \cos \phi=0\)

Current in a circuit when no power is dissipated is known as Watt less current.

Electrical Resonance

The current in a LCR circuit is given by

\(
I_0=\frac{V_0}{\sqrt{R^2+\left(X_C-X_L\right)^2}}=\frac{V_0}{Z}
Or, I_0=\frac{V_0}{\left.\sqrt{R^2+\left(\frac{1}{\omega C}-\omega L\right.}\right)^2}\)

Which indicates that at a particular angular frequency \(\omega_0, I_0\) will be maximum, Z or impendence of the circuit is minimum. The corresponding frequency is called resonant frequency.

At resonance, X_c = XL

\(\begin{aligned}
\omega_0 \mathrm{~L} & =\frac{1}{\omega_0 \mathrm{C}} \\
\omega_0^2 & =\frac{1}{\mathrm{LC}} \Rightarrow \omega_0=\frac{1}{\sqrt{\mathrm{LC}}} \\
∴ \mathrm{f}_0 & =\frac{1}{2 \pi \sqrt{\mathrm{LC}}}
\end{aligned}\)

• For a given value of L and C the resonant frequency does not depend on R.
• But the maximum current decreases with increase in the value of R.
• For smaller values of R the resonance curve is more sharp.
• The sharpness of the resonance curve is indicated by a term called quality factor Q.
• Quality factor is the ratio of the resonant frequency to the bandwidth.
• Bandwidth is the difference between half power frequencies.

\(\text { Quality factor }=\frac{\text { resonant frequency }}{\text { bandwidth }}\) \(i.e., Q=\frac{\omega_0}{2 \Delta \omega}
If \omega_0=2 \pi v_0, \omega_1=2 \pi v_1 and $\omega_2=2 \pi v_2\) \(Q=\frac{v_0}{v_2-v_1}\)

Quality factor is also given by,

\(Q=\frac{\omega_0}{2 \Delta \omega}=\frac{\omega_0 L}{R}\)

Since, \(\omega_0 L=\frac{1}{\omega_0 C}\) we can also write,

\(Q=\frac{1}{\omega_0 C R}=\frac{1}{R} \sqrt{\frac{L}{C}}\)

If Q value is less, sharpness is less. When sharpness is less, not only the maximum current is less, but also bandwidth is more and the tuning of the circuit will not be good.

If R is low or L is large, the Q is large and the circuit is more selective.

NEET Physics Electromagnetic Induction Notes

Electromagnetic Induction

The phenomenon of the production of emf across an electrical conductor in a changing magnetic field is called electromagnetic induction.

The magnetic flux through a surface is defined as,

\(\phi=\oint \overrightarrow{\mathrm{B}} \cdot \overrightarrow{\mathrm{A}}=\mathrm{BA} \cos \theta\) Where is the angle between a magnetic field and area vector.

Read And Learn More: NEET Physics Notes

Faraday’s Law of EMI

The magnitude of induced emf in a coil is equal to the rate of change of magnetic flux.

\(\text { i.e., } e=-\frac{d \phi}{d t}\)

If there are N turns in the coil,

\(e=-\frac{N d \phi}{d t}\)

The negative symbol indicates that the induced emf opposes the change in flux.

Lenz’s Law

The direction of induced emf or current in the circuit is such that, it opposes the cause that produced it.

Lenz’s law is in accordance with the law of conservation of energy.

Motional EMF

1. If a conducting rod of length is moving with a uniform velocity \(\overrightarrow{\mathrm{v}}\) perpendicular to a uniform magnetic field \overrightarrow{\mathrm{B}}, then the magnitude of induced emf in the rod is given by,
   \(\mathrm{e}-\mathrm{B} / \mathrm{v}\)
2. If a rod is moving by making an angle with the direction of a magnetic field, the
   \(\mathrm{e}=\mathrm{B} / \mathrm{v} \sin \theta\)
3. If a conductor starts sliding from the top of an inclined plane with the angle of inclination (θ) then,
   \(\begin{aligned}
   & \mathrm{e}=\mathrm{B} l v \sin (90-\theta) \\
   & \Rightarrow \mathrm{e}=\mathrm{B} l v \cos \theta
   \end{aligned}\)

Note:

Consider a conducting rod of length ‘l’ whose one end is fixed, is rotated about the axis passing through its fixed end and perpendicular to its length with constant angular velocity. The magnetic field ‘B’ is perpendicular to plane of the paper, then

The induced emf across the ends of the rod is given by,

\(\begin{aligned}
&E_{\text {sut }}=\frac{1}{2} \mathrm{~B} / v=\frac{\mathrm{B} l}{2}(\omega /)\\
&\mathrm{E}_{\mathrm{sa}}=\frac{1}{2} \mathrm{~B} \omega \ell^2
\end{aligned}\)

Where \(‘ \omega^{\prime}=2 \pi f\)

Self Induction

The phenomenon in which emf is induced in one coil due to a change in current in the same coil is called self-induction.

w.k.t., \(\phi \propto I \text { or } \phi=LI\)

Where L is called the coefficient of self-induction or self-inductance of the coil.

The induced emf in the coil is given by

\(\mathrm{e}=-\frac{\mathrm{d} \phi}{\mathrm{dt}} \Rightarrow \mathrm{e}=-\mathrm{L} \frac{\mathrm{dI}}{\mathrm{dt}}\) \(\text { If } \frac{\mathrm{dI}}{\mathrm{dt}}=1 \mathrm{As}^{-1} \text {, then }|\mathrm{e}|=\mathrm{L}\)

S.I. unit of L is Henry (H).

Dimensional formula of [L] = [ML² T^-2A^-2]

Self-inductance of a long solenoid

Where N is the total number of turns, A is the area of the Cross section, and ‘l’ is the length of the solenoid.

Mutual Induction

The phenomenon in which emf is induced in one coil due to a change in current in the neighboring coil is called mutual induction.

Induced emf due to mutual induction is given by,

\(\mathrm{E}=-\mathrm{M} \frac{\mathrm{dI}}{\mathrm{dt}}\)

Mutual Inductance of Two Long Co-Axial Solenoids

\(\mathrm{M}=\frac{\mu_0 \mathrm{~N}_1 \mathrm{~N}_2 \pi \pi_1^2}{l}\)

Where, l is the length of the solenoids, N₁ & N₂ are the number of turns in the inner and outer solenoids respectively is the radius of the inner solenoid.

Relation Between M, L₁ and L₂

For two magnetically coupled coils

\(\mathrm{M}=\mathrm{k} \sqrt{\mathrm{L}_1 \mathrm{~L}_2}\)

Where k is the coefficient of coupling or coupling factor.

\(\begin{gathered}
\mathbf{k}=\frac{\text { Magnetic flux linked in secondary }}{\text { Magentic flux linked in primary }} \\
0 \leq \mathrm{k} \leq 1
\end{gathered}\)

Combination of Inductors

If two coils of self inductances L₁ and L₂are in series and are far from each other, so that the mutual induction between them is negligible, then,

L_s = L₁ + L₂

If L₁ and L₂ are connected in parallel and if they are far from each other, then,

\(\frac{1}{L_p}=\frac{1}{L_1}+\frac{1}{L_2} \text { or } L_p=\frac{L_1 L_2}{L_1+L_2}\)

Induced emf in a coil is given by,

\(\begin{aligned}
& \mathrm{e}=\mathrm{NBA} \omega \sin \omega \mathrm{t} \\
& \mathrm{e}=\mathrm{e}_0 \sin \omega \mathrm{t}
\end{aligned}\)

Where, \(\mathrm{e}_0=\mathrm{NBA} \omega\) is the peak/maximum value of emf.

Choke coil

It is a device having high inductance and negligible resistance.

It consists of thick copper (Cu) wire to reduce the resistance of the circuit and a soft iron core to improve the inductance of the circuit.

Transformer works on the principle of mutual induction.

The efficiency of a transformer is given by

\(\eta=\frac{\text { Output power }}{\text { Input power }}=\frac{V_S I_S}{V_D I_D}\)

In a transformer,

\(\frac{N_s}{N_p}=\frac{V_S}{V_D}=\frac{I_p}{I_S}\)

\(\mathbf{k}=\frac{\mathrm{N}_{\mathrm{s}}}{\mathrm{N}_{\mathrm{p}}}\) is called turns ratio.

Energy stored in an inductor is given by,

\(\mathrm{U}=\frac{1}{2} \mathrm{LI}^2\)

Reasons for energy loss in a transformer:

1. Hysteresis loss.
2. Loss due to flux leakage
3. Loss due to resistance of the windings.
4. Loss due to eddy currents.

NEET Physics Magnetism and Matter Notes

Magnetism and Matter

Coulomb’s law in magnetism is given by,

\(\mathrm{F}=\frac{\mu_0}{4 \pi} \frac{\mathrm{q}_{\mathrm{x}} \mathrm{q}_{\mathrm{E}_1}}{\mathrm{r}^2}\)

Where is the permeability of free space, \(\mathrm{q}_{\mathrm{m}_1} \text { and } \mathrm{q}_{\mathrm{m}_2}\) are the pole strengths ‘r’ is the distance between two poles.

Read And Learn More: NEET Physics Notes

The magnetic moment of a magnet is given by,

\(\overrightarrow{\mathrm{m}}=\mathrm{q}_{\mathrm{m}}(2 \vec{\ell})\)

Where 2l is the separation between two magnetic poles.

The magnetic moment of a current-carrying loop is given by,

\(\overrightarrow{\mathrm{m}}=\mathrm{I} \overrightarrow{\mathrm{A}}\)

If there are N turns, \(\overrightarrow{\mathrm{m}}=\mathrm{NIA}\)

The magnetic moment of a revolving electron is given by,

\(\begin{gathered}
\mathrm{m}=\frac{\mathrm{eVT}}{2} \\
\mathrm{~m}=\frac{\mathrm{eVT}}{2} \times \frac{\mathrm{m}_{\mathrm{e}}}{\mathrm{m}_{\mathrm{e}}}=\frac{\mathrm{e}}{2 \mathrm{~m}_{\mathrm{e}}} \mathrm{L}
\end{gathered}\)

In vector form, \(\overrightarrow{\mathrm{m}}=-\left(\frac{\mathrm{e}}{2 \mathrm{~m}_{\ell}}\right) \overrightarrow{\mathrm{L}}\)

w.k.t., \(\mathrm{L}=\mathrm{n} \frac{\mathrm{h}}{2 \pi}\)

\(\mathrm{m}=\mathrm{n}\left(\frac{\mathrm{eh}}{4 \pi \mathrm{m}_{\mathrm{g}}}\right)\)

Bohr Magneton (m_B)

It is the magnetic dipole moment associated with an atom due to the orbital motion of an electron in the ground state of a hydrogen atom (n =1)

∴ \(\mathrm{m}_{\mathrm{B}}=\frac{\mathrm{eh}}{4 \pi \mathrm{m}_{\mathrm{e}}}=9.27 \times 10^{-24} \mathrm{Am}^2\)

The magnetic field on the axis of a bar magnet is given by:

\(\mathrm{B}_{\mathrm{xx}}=\frac{\mu_0}{4 \pi} \frac{2 \mathrm{mr}}{\left(\mathrm{r}^2-l^2\right)^2}\)

If r > > l then

\(B_{2 x}=\frac{\mu_0}{4 \pi} \frac{2 m}{r^3} \text { or } \vec{B}_{2 x}=\frac{\mu_0}{4 \pi} \frac{2 \vec{m}}{r^3}\)

Magnetic Field on the Equatorial Line of a Bar Magnet

\(\mathrm{B}_{\mathrm{*q}}=\frac{\mu_0 \mathrm{~m}}{4 \pi}\)

If r > >l then, \(B_{e q}=\frac{\mu_0}{4 \pi} \frac{m}{r^3}\)

In vector form,

\(\vec{B}_{s q}=-\frac{\mu_0 \vec{m}}{4 \pi r^3}\)

Magnetic field due to a bar magnet or magnetic dipole at a general point,

\(B_{\theta 4}=\frac{\mu_0}{4 \pi r^3} \sqrt{3 \cos ^2 \theta+1}\)

Torque experienced by a bar magnet when it is placed in a uniform magnetic field is given by,

\(\begin{gathered}
\vec{\tau}=\vec{m} \times \vec{B} \\
\tau=m B \sin \theta
\end{gathered}\)

The potential energy of a bar magnet placed in a uniform magnetic field.

\(\begin{aligned}
\mathrm{U} & =-\overrightarrow{\mathrm{m}} \cdot \overrightarrow{\mathrm{B}} \\
\mathrm{U} & =-\mathrm{mB} \cos \theta
\end{aligned}\)

Where is the angle between \(\overrightarrow{\mathrm{m}} \text { and } \overrightarrow{\mathrm{B}}\)

According to Gauss’ law in magnetism, the surface integral of magnetic field over a closed surface is always zero.

\(\text { i.e., } \oint_{\mathrm{s}} \overrightarrow{\mathrm{B}} \cdot \overrightarrow{\mathrm{ds}}=0\)

Consequences of Gauss’ Law in Magnetism

Number of magnetic field lines leaving the surface is equal to number of magnetic field lines entering the surface.

Magnetic monopoles do not exist i.e., poles exist in unlike pairs of equal strengths.

The magnetic moment induced per unit volume of the substance is the intensity of magnetization.

\(I=\frac{M}{V} A m^{-1}\)

Magnetic Susceptibility(χ):

\(\begin{gathered}
I \propto H \\
I=\chi H \\
\text { or } \chi=\frac{I}{H}
\end{gathered}\)

Magnetic Permeability(μ):

\(\mu=\frac{B}{H}\)

Relative magnetic permeability

\(\mu_r=\frac{\mu}{\mu_0}\)

Relations amongst B, H, I, \(\chi \text { and } \mu\)

\((1) B=\mu H=B_0+\mu_0 I=\mu_0(H+I)
(2) B=\mu_0 H(1+\chi)
(3) \mu=\mu_0(1+\chi)
(4) \mu_{,}=(1+\chi)\)

Magnetic Hysteresis Loop, B-H or I-H Curve

1. B always lags behind H. This property is called magnetic hysteresis.
2. The curve drawn between B and H is called B-H curve. It is a closed curve for one complete cycle of magnetization.
3. Hysteresis loss per cycle per unit volume = area bounded by the hysteresis loop.

Types of Magnetic Materials

1. Diamagnetic materials: They are weakly repelled by the bar magnet.
2. Paramagnetic materials: They are weakly attracted by the bar magnet.
3. Ferromagnetic materials: They are strongly attracted by the magnet

Curie’s Law and Curie Temperature

For a paramagnetic material, magnetic susceptibility is inversely proportional to absolute temperature.

\(\text { i.e., } \chi \propto \frac{1}{T} \quad \text { or } \chi=\frac{C}{T}\)

For a ferromagnetic material

\(\chi=\frac{C}{\left(T-T_C\right)}\)

where T_c is called curie temperature.

The temperature at or above which the ferromagnetic material behaves like a paramagnetic material is called Curie temperature.

Earth’s Magnetism

It is described in terms of three quantities, which are called magnetic elements of the earth.

These are:

Magnetic declination (000) at a place is the angle between the magnetic meridian and the geographic meridian.

Magnetic Inclination or dip at a place is defined as the angle , which is the direction of total intensity of earth’s magnetic field, B makes with a horizontal line in magnetic meridian.

Horizontal component, H is the component of total intensity of earth’s magnetic field (B) in the horizontal direction in magnetic meridian,

\(\begin{aligned}
& \mathrm{H}=\mathrm{B} \cos \delta \\
& \mathrm{V}=\mathrm{B} \sin \delta
\end{aligned}\)

∴ \(\sqrt{H^2+V^2}=B\)

and \(\frac{V}{H}=\tan \delta\)

NEET Physics Moving Charges and Magnetism Notes

Moving Charges and Magnetism

The force experienced by a charged particle when it is moving in a uniform magnetic field is

1. Directly proportional to the strength of the magnetic field. i.e., \(F \propto B\)
2. Directly proportional to the magnitude of charge. i.e., \(F \propto q\)
3. Directly proportional to the component of velocity in a direction perpendicular to the direction of the magnetic field. i.e., \(F \propto v\) sinθ

Read And Learn More: NEET Physics Notes

∴\(\begin{aligned}
& \mathrm{F} \propto \mathrm{qvB} \sin \theta \\
& \mathrm{F}=\mathrm{kqvB} \sin \theta
\end{aligned}\)

In SI the value of k = 1

\(\mathrm{F}=\mathrm{qvB} \sin \theta\)

In vector form, \(\overrightarrow{\mathrm{F}}=\mathrm{q}(\overrightarrow{\mathrm{v}} \times \overrightarrow{\mathrm{B}})\)

Case 1:

The magnetic field on q is zero if it is at rest (since v=0)

Case 2:

If angle between \(\overrightarrow{\mathrm{v}} \text { and } \overrightarrow{\mathrm{B}} \text { is } 0^{\circ} \text { or } 180^{\circ}\) is 0º or 180º

Then, F = 0 (∵sin 0º = 0 and sin 180º = 0)

Case 3:

When the charged particle moves perpendicular to the direction of magnetic field, then,

F = qvB(1) (∵sin 90º = 1)

\(\mathrm{F}_{\max }=\mathrm{qvB}\)

SI unit of the magnetic field is tesla (T)

1 tesla (T) = 1 Weber meter^-2 (Wbm^-2)

Note:

The magnetic field is commonly expressed in gauss (G).

1 gauss (G) = 10^-4 tesla(T)

1G = 10^-4 T

Lorentz Force

If a charged particle moves in a region where both electric and magnetic fields are present, then the force experienced by the charged particle is called Lorentz force.

\(\begin{aligned}
& \vec{F}_L=\vec{F}_E+\vec{F}_B \\
& \vec{F}_L=q \vec{E}+q(\vec{v} \times \vec{B})=q[\vec{E}+(\vec{v} \times \vec{B})]
\end{aligned}\)

Motion of a charged particle inside a uniform magnetic field

1. When is perpendicular to \(\overrightarrow{\mathbf{B}}\):

Using Flemings left hand rule it can be shown that the force acting on the charged particle is centripetal.

i.e., the necessary centripetal force for the revolution is given by magnetic force.

Where ‘r’ is the radius of the circular path.

The time period of revolution of a charged particle is given by,

\(\begin{aligned}
&\mathrm{T}=\frac{\text { circumference }}{\text { speed }}=\frac{2 \pi \mathrm{r}}{\mathrm{v}}=\frac{2 \pi}{\mathrm{v}}\left(\frac{\mathrm{mv}}{\mathrm{qB}}\right)\\
&\mathrm{T}=\frac{2 \pi \mathrm{m}}{\mathrm{qB}}
\end{aligned}\)

The frequency of revolution is given by

\(\mathrm{f}=\frac{\mathrm{qB}}{2 \pi \mathrm{m}}\)

Note:

Time period and frequency are independent of velocity of the charged particle and radius of the circular path.

If the velocity of the charged particle increases, then the radius of the path also increases, without any change in time period, and frequency.

2. When and are inclined to each other, the charged particle moves in a helical path:

The distance traveled by the charged particle in the direction of the magnetic field in a time equal to the time period is called pitch of the helical path.

\(\text { pitch }=v \cos \theta \times \frac{2 \pi m}{q B}\)

Cyclotron

It is a device used to accelerate heavy-charged particles.

Maximum velocity of the charged particle in a cyclotron is given by,

\(\mathrm{v}_{\max }=\frac{\mathrm{qBR}}{\mathrm{m}}\)

Where ‘R’ is the radius of the dees.

The maximum kinetic energy gained is given by,

\(\begin{aligned}
& \mathrm{E}_{\max }=\frac{1}{2} \mathrm{mv}_{\max }^2=\frac{1}{2} \mathrm{~m}\left(\frac{\mathrm{qBR}}{\mathrm{m}}\right)^2 \\
& \mathrm{E}_{\max }=\frac{1}{2} \frac{\mathrm{q}^2 \mathrm{~B}^2 \mathrm{R}^2}{\mathrm{~m}}
\end{aligned}\)

Note:

Cyclotron is not suitable for accelerating electrons.

Since the mass of the electron is very small its speed increases rapidly and starts moving at a relativistic speed.

The magnetic field at a point due to a current-carrying conductor can be calculated using Biot–Savart’s law.

According to Biot–Savart’s law the magnetic field dB produced by a small conductor of length d and carrying current I is given by,

\(\mathrm{dB}=\frac{\mu_0}{4 \pi} \frac{\mathrm{Id} \ell \sin \theta}{\mathrm{r}^2}\)

Where ‘r’ is the distance between the current element and the point where we want to calculate the magnetic field and θ is the angle between the direction of the current element and the line joining the current element and the point of observation.

The magnetic field at a point on the axis of a circular loop carrying current

Where N is the number of turns in the coil.

x is the distance between the point of observation and the center of the loop.

Case 1 :

Magnetic field at the center of the loop (x = 0)

\(\begin{aligned}
& \mathrm{B}=\frac{\mu_0}{4 \pi} \frac{2 \pi \mathrm{NI}}{\mathrm{R}} \\
& \mathrm{B}=\frac{\mu_0 \mathrm{NI}}{2 \mathrm{R}}
\end{aligned}\) \(\text { If } \mathrm{N}=1 \text {, then, } \mathrm{B}=\frac{\mu_0 I}{2 \mathrm{R}}\)

Case 2:

When x >> R,

\(\begin{aligned}
& \mathrm{B}=\frac{\mu_0}{4 \pi} \frac{2 \pi \mathrm{NIR}^2}{\mathrm{x}^3} \\
& \mathrm{~B}=\frac{\mu_0}{4 \pi} \frac{2 \mathrm{~m}}{\mathrm{x}^3} \quad(∵\mathrm{m}=\mathrm{NIA})
\end{aligned}\)

Where ‘m’ is known as magnetic moment of the coil.

Ampere’s Circuital Law

It states that “the line integral of magnetic field around a closed loop enclosing an arbitrary area is equal to times the total current passing through the area”.

\(\text { Mathematically, } \oint \overrightarrow{\mathrm{B}} \cdot \mathrm{d} \vec{\ell}=\mu_0 \mathrm{I}\)

Magnetic field due to a long current carrying wire is given by,

Where ‘r’ is the perpendicular distance between the point of observation and the wire.

Magnetic field due to a current carrying wire of finite length is given by

Where, φ1and φ2 are the angles, which the lines joining the observation point with the two ends of the conductor make with the normal to the conductor from the observation point.

The magnetic field due to a long straight solenoid is given by,

\(\mathrm{B}=\mu_0 \mathrm{nI}\)

Where ‘n’ is the number of turns per unit length.

The magnetic field produced by a toroid is given by,

\(\mathrm{B}=\mu_0 \mathrm{nI}\)

The magnetic field due to a circular arc of wire subtending an angle θ at the center

If (for 1 complete loop)

\(\mathrm{B}=\frac{\mu_0 \mathrm{I}}{2 \mathrm{r}}\)

Magnetic force on a current-carrying conductor is given by,

\(\mathrm{F}=\mathrm{BI} l \sin \theta\)

In vector form, \(\overrightarrow{\mathrm{F}}=\mathrm{I}(\vec{\ell} \times \overrightarrow{\mathrm{B}})\)

The force between two infinitely long parallel current-carrying conductors is:

\(\mathrm{F}=\frac{\mu_0}{4 \pi} \frac{2 \mathrm{I}_1 \mathrm{I}_2}{\mathrm{r}}\)

∴ The force per unit length is given by,

\(\mathrm{f}=\frac{\mathrm{F}}{l}=\frac{\mu_0}{4 \pi} \frac{2 \mathrm{I}_1 \mathrm{I}_2}{\mathrm{r}}\)

Definition of Ampere

Let I1 = I2 = 1A, r = 1m, then

\(\mathrm{f}=10^{-7} \times \frac{2 \times 1 \times 1}{\mathrm{r}}=2 \times 10^{-7} \mathrm{~N}\)

Thus, one ampere is that current, which when flowing through each of the two parallel conductors of infinite length and placed in free space at a distance of 1 metre from each other, produces between them a force of newton per meter of their lengths.

The torque acting on a current loop placed in a magnetic field:

\(\tau=\text { BIA } \sin \theta\)

If there are N loops, then

\(\tau=\text { NBLA } \sin \theta
Or
\tau=\mathrm{mB} \sin \theta \quad(∵\mathrm{m}=\mathrm{NIA})\)

Mathematically, \(\vec{\tau}=\vec{m} \times \vec{B}\)

Figure of merit of a galvanometer is the current required to produce unit deflection in the galvanometer

The current sensitivity of a galvanometer is given by,

\(I_S=\frac{\theta}{I}=\frac{N B A}{K}\)

The voltage sensitivity of a galvanometer is given by,

\(\mathrm{V}_{\mathrm{S}}=\frac{\theta}{\mathrm{V}}=\frac{\theta}{\mathrm{IR}}=\frac{\mathrm{NBA}}{\mathrm{KR}}\)

Conversion of galvanometer into an ammeter

A galvanometer can be converted into an ammeter by connecting a low resistance called a shunt in parallel with the galvanometer coil.

\(S=\frac{I_g G}{I-I_g}\)

The resistance of the ideal ammeter is zero.

Conversion of a galvanometer into a voltmeter

This can be done by connecting a high-resistance R in series with the galvanometer.

\(R=\frac{V}{I_g}-G\)

Resistance of a ideal voltmeter is infinity.