First slide

Magnetic force

Question

A square loop of edge ‘l’ and carrying a current $i$ , is placed with its edges parallel to the $X Y$ axes. There is a non-uniform magnetic field present in the region $\vec{B} = B_{0} (1 + \frac{x}{l}) (- \hat{k})$ . Find the magnitude of the net magnetic force experienced by the loop.

Moderate

A

$i B_{0} l$
B

$\frac{i B_{0} l}{2}$
C

$\frac{i B_{0} l}{3}$
D

$\frac{i B_{0} l}{4}$

Solution

The magnetic field in the region
$B = B_{0} (1 + \frac{x}{l})$
Force on AB (along -x axis),
$F_{1} = i B_{0} (1 + \frac{0}{l}) = i B_{o} l$
Force on CD (along +x axis),
$F_{2} = i B_{0} (1 + \frac{l}{l}) l = i B_{o} .2 l$
The force due to wires BC (along +y axis) and AD (along -y axis) will be equal and opposite. Hence, the force on BC and AD will cancel out each other. Net force, $F_{2} - F_{1} = i B_{0} l$ . (along +x axis)

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