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:

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

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

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 Alternating Current Notes