RC circuits: A 6.0-μF capacitor is connected in series with a 5.0 MΩ resistor, and this combination is connected across an ideal 15-V DC battery. What is the current in the circuit when the capacitor has reached 20% of its maximum charge?
A) 6.5 μA
B) 2.4 μA
C) 1.3 μA
D) 4.7 μA
E) 9.1 μA

Question

Quizplus · Accepted Answer

When the capacitor has reached 20% of its maximum charge, we know that the voltage across the capacitor is 0.2 times the total voltage of the battery, or 3 V. We can use the formula for the charging of a capacitor in an RC circuit:

V = V0(1 - e^(-t/RC))

where V0 is the voltage of the battery, t is the time, R is the resistance, and C is the capacitance. We can rearrange this equation to solve for the time:

t = -RC ln(1 - V/V0)

Plugging in the given values, we get:

t = -(5.0 x 10^6 Ω)(6.0 x 10^-6 F) ln(1 - 3/15)

t = 1.85 x 10^6 s

Now that we have the time, we can use the formula for the current in the circuit:

I = V/R e^(-t/RC)

Plugging in the given values, we get:

I = (15 V)/(5.0 x 10^6 Ω) e^(-1.85 x 10^6 s/(5.0 x 10^6 Ω)(6.0 x 10^-6 F))

I ≈ 2.4 μA

Therefore, the best choice is B.

RC Circuits: a 6