First slide

Force on a current carrying wire placed in a magnetic field

Question

A long straight wire carrying current I₁ is placed in the plane of a ribbon carrying current I₂ parallel to the wire. The width of ribbon is b. The straight conductor is placed at a distance a from the nearer edge of ribbon. Find the force of attraction per unit length between the two.

Difficult

A

$\frac{μ_{0} I_{1} I_{2}}{2 π} \log \frac{a + b}{a}$
B

$\frac{μ_{0} I_{1} I_{2}}{2 π} \log \frac{a}{a + b}$
C

$\frac{μ_{0} I_{1} I_{2}}{2 π} \log \frac{a}{b}$
D

$\frac{μ_{0} I_{1} I_{2}}{2 π} \log \frac{(a + b)}{a}$

Solution

Consider an elementary strip of width dx on the sheet at a distance x from the wire.
Force on this element is, $dF = \frac{μ_{0} I_{1}}{2 πx} I_{2} dx$
Total force on unit length of sheet is
$\begin{array}{l} F = \int_{a}^{a + b} \frac{μ_{0} I_{1}}{2 πx} I_{2} dx \\ F = \frac{μ_{0} I_{1} I_{2}}{2 π} \int_{a}^{a + b} \frac{1}{x} dx = \frac{μ_{0} I_{1} I_{2}}{2 π} \log \frac{(a + b)}{b} \end{array}$

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