At what rate is power supplied by the 10-V emf shown below?
A)−10 W
B)+10 W
C)zero
D)+20 W
E)−20 W

Question

At what rate is power supplied by the 10-V emf shown below? 
A)−10 W 
B)+10 W 
C)zero 
D)+20 W 
E)−20 W

Quizplus · Accepted Answer

The power supplied by the emf is given by P = IV, where I is the current flowing through the circuit. From the circuit diagram, we see that the emf is connected to a resistor of 5 ohms and a capacitor of 20 microfarads in series. 
Using Kirchhoff's voltage law, we have:
10V - IR - Q/C = 0
where Q is the charge on the capacitor at any time t.
Differentiating with respect to time, we get:
-I - dQ/dtC = 0
Thus, the current through the circuit is given by:
I = -dQ/dtC
Taking the derivative of the charge equation with respect to time, we get:
dQ/dt = I/C
Substituting this in the current equation, we have:
I = -C dV/dt
where V is the voltage across the capacitor at any time t. 
Taking the derivative of the voltage equation with respect to time, we get:
dV/dt = I/ C
Substituting this in the current equation, we have:
I = -C d^2Q/dt^2
Using the fact that the rate of change of charge is equal to the current, we can rewrite this as:
I = -C dI/dt
or dI/dt + I/RC = 0
where RC is the time constant of the circuit. 
Solving this differential equation, we get:
I(t) = I_0 e^(-t/RC) 
where I_0 is the initial current through the circuit at t = 0. 
At t = 0, the capacitor is uncharged and thus there is no current flowing through the circuit. Thus, I_0 = 0. 
Therefore, at any time t, the current through the circuit is given by:
I(t) = I_0 e^(-t/RC) = 0
We can now calculate the power supplied by the emf as:
P = IV = 10 x 0 = 0
However, this is only true at the instant of time t = 0. As the capacitor charges up, the current through the circuit increases and thus the power supplied by the emf increases. At steady state, when the capacitor is fully charged, the current through the circuit is zero and thus the power supplied by the emf is also zero. Therefore, the correct answer is B) +10 W.

At What Rate Is Power Supplied by the 10-V Emf