Set 52 Problem number 4

Problem

A battery with negligible internal resistance maintains a potential difference of 8 volts in a uniform conducting wire 5 meters long.  What force is experienced by a conduction electron in this wire (electron charge has a magnitude of 1.6 * 10^-19 Coulomb)?

Solution

From the charge (Coulombs) and voltage (Joules / Coulomb) we can find the amount of work done on a given charge over the length of the wire. Then from the work and distance we can easily find the force.

• The work done is 8 J/C (1.6 x 10^-19 C) = 12.8 x 10^-19 J = 12.8 x 10^-19 N m.
• If 12.8 x 10^-19 N m of work are done by a uniform force over a distance of 5 meters, the force is 12.8 x 10^-19 N m / ( 5 m) = 2.56 * 10^-19 N.

Generalized Solution

The work done on charge q through potential difference `dV is `dW = q `dV.

• If this work is done over distance `dx, then we see that the average force exerted on q is Fave = `dW / `dx = q `dV / `dx.
• The quantity `dV / `dx is the average potential gradient, also called the average electric field, over the path followed by the charge.

Note that the force on the charge is always in the direction of the wire, as is the distance `dx.