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AC Circuits — Physics STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 SciencePhysicsAC Circuits4 Marks
Question
An alternating emf is applied to a CR circuit. Obtain an expression for the phase difference between the emf and the current. Also obtain the expression for the effective resistance of the cir-cuit. Assume the capacitor and resistor to be ideal. Draw the phasor diagram showing the emf and current.
✓

Answer

Below figure shows a source of alternating emf (e), key K, ideal capacitor of capacitance C and ideal resistor of resistance R to form a closed series circuit. Ignoring the resistance of the source and the key, we have,
Image
where $C$ is the time independent constant of integration which must be zero as $j$ oscillates about zero when e oscillates about zero.
$\therefore e = Ri _{ O } \sin \omega t -\frac{i_0}{\omega C} \cos \omega t$
Let $Z =\sqrt{R^2+\frac{1}{\omega^2 C^2}}, R = Z \cos$ and $\frac{1}{\omega C}= Z \sin \Phi$
$\therefore e=i_0 Z(\cos \Phi \sin \omega t-\sin \Phi \cos \omega t)$
$=Z_{i 0}(\sin \omega t \cos \Phi-\cos \omega t \sin \Phi)$
$=Z i_0 \sin (\omega t-\Phi)=e_0 \sin (\omega t-\Phi)$, where $e_0=Z i_0$ is the peak emf. Here, the emf lags behind the current by phase angle $\Phi$.
Image

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