Question

A 2.00 nF capacitor with an initial charge of 5.01 µC is discharged through a 2.62 k resistor.

(a) Calculate the current in the resistor 9.00µs after the resistor is connected across the terminals of the capacitor.
mA
(b) What charge remains on the capacitor after 8.00 µs?
µC
(c) What is the maximum current in the resistor?
A

Answer 1

Concepts and reason

The concept used to solve the problem is that the current through a resistor and the charge on a capacitor decrease exponentially with time when a charged capacitor discharges through a resistor. The time constant of the RC circuit is determined and the value is used in the expressions for the exponential decay of current and the charge.

Fundamentals

When a charged capacitor is connected to a resistor, the capacitor discharges through the resistor. The potential difference across the resistor before discharge is equal to the potential difference across the capacitor. When the capacitor discharges, the current I flowing through the resistor at an instant of time t is given by the expression,

…… (1)

Here, the maximum current flowing through the resistor is , and is the time constant.

The maximum current is given by,

…… (2)

Here is the initial charge on the plates of the capacitor.

The time constant is given by,

……(3)

Here, R is the resistance of the resistor and C is the capacitance of the capacitor.

The charge Q that remains on the plates of the capacitor after a time t is given by the expression,

……(4)

(c)

A capacitor of capacitance C is charged so that a charge resides on its plates. This is connected to a resistor of resistance R. The capacitor discharges through the resistor, causing a current to flow through it. As the capacitor discharges, the potential difference across the resistor decreases and hence the current decreases. Therefore, the maximum current flows through it at time

Express the capacitance C in F.

Express the resistance R in .

Calculate the time constant of the RC circuit by substituting for R and for C in equation (3).

Express the time constant in .

Calculate the maximum current by substituting for and for in equation (2).

Express the current in mA.

(a)

The current I, at any instant of time, flowing through the resistor, is given by equation (1).

Substitute 956 mA for , for t and for in equation (1).

(b)

The capacitor, which is charged so that it initially contains an amount of charge , when allowed to discharge through a resistor, the charge on the plates of the capacitor is found to decrease exponentially, as is given by equation (4).

Substitute for , for t and for in equation (4).

Ans: Part c

The maximum current flowing through the resistor when the capacitor discharges through it is 956 mA.

Part a

The current flowing in the resistor 9.00 µs after the discharge commences is 172 mA.

Part b

The charge on the plates of the capacitor 8.00 µs after the commencement of the discharge is 1.09 µC.