H2 Physics 15 - Currents: Drift Velocity in a Current-Carrying Wire

Step 1 - State the problem and given data

A copper wire of cross-sectional area one point five times ten to the minus six square metres carries a current of three point zero amperes.

Step 1 - State the problem and given data

The number density of free electrons in copper is eight point five times ten to the twenty-eight per cubic metre.

Step 1 - State the problem and given data

We want to find the drift velocity of the electrons.

Step 2 - Write the current - drift velocity relation

The current I in a conductor is related to drift velocity by I equals n A v e, where n is the number density of charge carriers, A is the cross-sectional area, v is the drift velocity, and e is the charge on each carrier.

Step 2 - Write the current - drift velocity relation

Rearranging for v gives v equals I divided by n A e.

Step 3 - Substitute the values

Substituting: v equals three point zero divided by the product of eight point five times ten to the twenty-eight, one point five times ten to the minus six, and one point six zero times ten to the minus nineteen.

Step 3 - Substitute the values

Let us compute the denominator step by step.

Step 4 - Evaluate the denominator and find v

First, eight point five times one point five gives twelve point seven five.

Step 4 - Evaluate the denominator and find v

Then twelve point seven five times one point six zero gives twenty point four.

Step 4 - Evaluate the denominator and find v

The powers of ten: ten to the twenty-eight times ten to the minus six times ten to the minus nineteen gives ten to the three.

Step 4 - Evaluate the denominator and find v

So the denominator is twenty point four times ten to the three, which is two point zero four times ten to the four.

Step 4 - Evaluate the denominator and find v

Therefore v equals three point zero divided by two point zero four times ten to the four, giving approximately one point five times ten to the minus four metres per second.

Step 5 - Physical interpretation

The drift velocity is remarkably small, just zero point one five millimetres per second.

Step 5 - Physical interpretation

This is because there are an enormous number of free electrons in copper, so each one only needs to drift very slowly for the total current to reach three amperes.

Step 5 - Physical interpretation

A common exam question asks: if the wire narrows, what happens to drift velocity?

Step 5 - Physical interpretation

Since I equals n A v e and the current is the same throughout a series circuit, a smaller cross-sectional area means a larger drift velocity. The electrons speed up in the thinner section.