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Physics · 2023 · 3 marks
CBSE 2023 · Region 3 · Set 1 · Q28
(a) An ac source v = v_m sin ω t is connected across an ideal capacitor. Derive the expression for the (i) current flowing in the circuit, and (ii)…
(a)
An ac source $\displaystyle \mathrm{v}=\mathrm{v}_{\mathrm{m}} \sin \omega \mathrm{t}$ is connected across an ideal capacitor. Derive the expression for the (i) current flowing in the circuit, and
(ii)
reactance of the capacitor. Plot a graph of current i versus $\displaystyle \omega \mathrm{t}$.
OR
(b)
A series combination of an inductor L , a capacitor C and a resistor $\displaystyle R$ is connected across an ac source of voltage in a circuit. Obtain an expression for the average power consumed by the circuit. Find power factor for (i) purely inductive circuit, and (ii) purely resistive circuit.
Marking-scheme solution
(i)
\(\displaystyle V_{\mathrm{m}} \operatorname{Sin} \omega \mathrm{t}=\frac{q}{C}\)
\[I=\frac{d q}{d \mathrm{t}}=\frac{d}{d \mathrm{t}}\left(C V_{\mathrm{m}} \sin \omega \mathrm{t}\right)
\]
\(\displaystyle I=\omega C V_{\mathrm{m}} c \cos \omega \mathrm{t}\)
Alternatively:-
\[I=\frac{V_{\mathrm{m}}}{1 / \omega C} \cos \omega \mathrm{t}=I_{\mathrm{m}} \sin (\omega \mathrm{t}+\pi / 2)
\]
(ii)
\(\displaystyle I=\frac{V_{\mathrm{m}}}{1 / \omega C} \sin (\omega \mathrm{t}+\pi / 2)=I_{\mathrm{m}} \sin (\omega \mathrm{t}+\pi / 2)\)
Comparing with \(\displaystyle I_{\mathrm{m}}=\frac{V_{\mathrm{m}}}{1 / \omega C}\)
Reactance of the capacitor; \(\displaystyle X_{c}=\frac{1}{\omega C}\)
Instantaneous Power;
\(\displaystyle P=V I=\left(V_{m} \sin \omega t\right) \times I_{m} \sin (\omega t+\phi)\)
\[P=\frac{V_{m} I_{m}}{2}[\cos \phi-\cos (2 \omega t+\phi)] \quad \text{----(1)}
\]
The average power over a cycle is given by the average of the two terms in the R.H.S of equation ($\displaystyle 1$). It is only the second term which is time dependent. Its average is zero (the positive half of the cosine cancels the negative half). Therefore,
\[P_{avg}=\frac{V_{m} I_{m}}{2} \cos \phi=\frac{V_{m}}{\sqrt{2}} \frac{I_{m}}{\sqrt{2}} \cos \phi
\]
\(\displaystyle P_{avg}=V_{rms} I_{rms} \cos \phi\)
(i)
Power factor for purely inductive circuit, \(\displaystyle \phi=\frac{\pi}{2} \Rightarrow \cos \phi=0\)
(ii)
Power factor for purely resistive circuit; \(\displaystyle \phi=0 \Rightarrow \cos \phi=1\)
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