First slide

Force on a current carrying wire placed in a magnetic field

Question

The magnetic field existing in a region is given by $\vec{B} = B_{0} (1 + \frac{x}{l}) \hat{k} .$ A square loop of edge l and carrying current I is placed with its edges parallel to the x and y axes. The magnitude of the net magnetic force experienced by the loop is

Moderate

A

$2 B_{0} Il$
B

$B_{0} Il$
C

$\frac{1}{2} B_{0} Il$
D

$4 B_{0} Il$

Solution

$\vec{F} = B_{0} i l [(1 + \frac{a_{0}}{l}) (- \hat{i}) + (1 + \frac{a_{0} + l}{l}) (\hat{i})]$
Magnetic force on ab and cd are equal and opposite
$∴ {\vec{F}}_{net} = {\vec{F}}_{1} + {\vec{F}}_{2} = ({ilB}_{0}) \hat{i}$

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