Potential Difference Between Two Points — Question 8

Question 8

Potential gradients inside conducting wires reconcile nonzero current with small voltage drop

A straight uniform wire of length L, area A and resistivity rho carries steady current I toward +x. Let V(0)=V0 and assume local Ohm behavior.

1. Find V(x) and its electric field.

2. For two taps separated by distance ell, find their voltage difference; explain the ideal-wire limit at fixed current.

Question 8 - Solution

Potential gradients inside conducting wires reconcile nonzero current with small voltage drop

(a) Method and calculation

$$J=I/A,\quad E_x=\rho J=\rho I/A,\quad dV/dx=-E_x.$$$$V(x)=V_0-\rho Ix/A.$$

(b) Interpretation and further calculation

$$V(x+\ell)-V(x)=-\rho I\ell/A.$$$$\rho\to0\text{ at fixed }I:\quad E_x\to0,\quad\Delta V\to0.$$

Physical check

A real resistive wire needs a small field for steady current; an ideal zero-resistance wire is a limiting model.