What are the corresponding exponential decay equations for the current and the potential difference during discharge?

They are I = I₀e⁻ᵗ/ᴿᶜ and V = V₀e⁻ᵗ/ᴿᶜ, with the same time constant RC.

What linear equation is obtained by taking natural logarithms of Q = Q₀e⁻ᵗ/ᴿᶜ?

Taking natural logarithms of both sides gives ln Q = ln Q₀ − t/RC, which is of the form y = c + mx.

Which graph would you plot to obtain a straight line from discharge data, and what would its gradient be?

A graph of ln V against t gives a straight line of gradient −1/RC with intercept ln V₀.

How is the time constant obtained from the gradient of a ln V against t graph?

The time constant is found from RC = −1 divided by the gradient.

Why does a discharging capacitor never quite reach zero charge in theory?

Because the decay is exponential, so the charge falls by the same fraction in each equal time interval and only approaches zero asymptotically.

Roughly what fraction of the original charge remains after five time constants?

Less than one per cent remains, so the capacitor is treated as fully discharged.

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Electric and Magnetic Fields

The exponential decay equations and logarithmic graphs

1 thing to know
Papers 1 and 3

Overview

Recall questions on the decay equations for charge, current and potential difference, and the linear form obtained by taking natural logarithms.

What you need to know

Answer each one before you reveal it.

Your recallNot started

Numbers

Which graph gives a straight line from discharge data, how is the time constant found from it, and why is zero never reached?

Questions from this subtopic

Each one with the model answer underneath.

1
What are the corresponding exponential decay equations for the current and the potential difference during discharge?
Model answer
They are I = I₀e⁻ᵗ/ᴿᶜ and V = V₀e⁻ᵗ/ᴿᶜ, with the same time constant RC.
2
What linear equation is obtained by taking natural logarithms of Q = Q₀e⁻ᵗ/ᴿᶜ?
Model answer
Taking natural logarithms of both sides gives ln Q = ln Q₀ − t/RC, which is of the form y = c + mx.
3
Which graph would you plot to obtain a straight line from discharge data, and what would its gradient be?
Model answer
A graph of ln V against t gives a straight line of gradient −1/RC with intercept ln V₀.
4
How is the time constant obtained from the gradient of a ln V against t graph?
Model answer
The time constant is found from RC = −1 divided by the gradient.
5
Why does a discharging capacitor never quite reach zero charge in theory?
Model answer
Because the decay is exponential, so the charge falls by the same fraction in each equal time interval and only approaches zero asymptotically.
6
Roughly what fraction of the original charge remains after five time constants?
Model answer
Less than one per cent remains, so the capacitor is treated as fully discharged.

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