Question

If a battery has been connected to the circuit for a long time, then the capacitor discharges half of this stored energy in exactly

a) 1 time constant
b) 2 time constants
c) 3 time constants
d) 4 time constants

Answer 1

The time it takes for a capacitor to discharge to half its initial energy is determined by the relationship between energy and voltage (E = 1/2 CV^2) and occurs after approximately 0.693 time constants. This does not match any of the given options.

Step-by-step explanation:

The question pertains to the concept of an RC time constant in a circuit with a capacitor and concerns the discharge rate of the capacitor.

For a capacitor discharging through a resistor, the voltage across the capacitor and the energy stored in the capacitor decrease exponentially. The amount of time it takes for a capacitor to discharge to half of its energy value is not directly given by an RC time constant. Instead, we use the relationship that the energy stored in a capacitor is proportional to the square of the voltage across the capacitor (E = 1/2 CV2).

Thus, to discharge to half of its initial energy, the voltage across the capacitor must drop to 1/√2 (about 70.7%) of its initial value. This occurs after approximately 0.693 time constants, as determined by the natural logarithm of 2 (ln(2)). Therefore, none of the provided options (a, b, c, d) accurately describe the time it takes for a capacitor to discharge half of its stored energy.