A capacitor of capacitance \(C\) is connected to a battery of emf \(\varepsilon\) at \(t = 0\) through a resistance \(R\). Find the maximum rate at which energy is stored in the capacitor. When does the rate has this maximum value ?
1. \(\frac{\varepsilon^2}{4R}\)
2. \(\frac{\varepsilon^2}{2R}\)
3. \(RC\)
4. \(CR \ln 2\)
View Answer
The energy stored rate is \(P = \frac{\varepsilon^2}{R} e^{-t/RC}(1-e^{-t/RC})\). This is maximum when \(e^{-t/RC} = \frac{1}{2}\). The maximum rate is \(P_{max} = \frac{\varepsilon^2}{4R}\). This occurs at \(t = RC \ln 2\).
Assertion (A): Capacitor reduces sparks in induction coil.
Reason (R): Capacitor provides alternative path to current when circuit is broken.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
A: True. In an induction coil, breaking the circuit induces a high back EMF, causing sparks across the switch. Capacitors are used to mitigate this.
R: True. A capacitor connected across the switch provides a path for the induced current, absorbing the inductive energy and preventing excessive voltage buildup that leads to sparks.\n(R) correctly explains how (A) works.