Chapter 29 Problem 35

Problem

A short section of wire, of length $a$, is moving with velocity ${\vec {v}}$ , parallel to a very long wire carrying a current $I$. The near end of the wire section is a distance $b$ from the long wire. Assuming the vertical wire is very long compared to $a+b$, determine the emf between the ends of the short section. Assume ${\vec {v}}$ is

(a) in the same direction as $I$,

(b) in the opposite direction to $I$.

Solution

The magnetic field is perpendicular to the rod, with the magnetic field decreasing with distance from the rod.

(a)

$d{\mathcal {E}}=Bvdr$

$B={\frac {\mu _{0}}{2\pi }}{\frac {I}{r}}$ (near a straight long wire)

${\mathcal {E}}=\int d{\mathcal {E}}=\int _{b}^{b+a}{\frac {\mu _{0}I}{2\pi r}}vdr={\frac {\mu _{0}Iv}{2\pi }}\ln \left({\frac {b+a}{b}}\right)$

The force is towards the long wire

(b)

same magnitude, away from the long wire