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Q.

A circular loop of radius a, carrying a current $i$ , is placed in a two dimensional magnetic field. The center of the loop coincides with the center of the field as shown in Fig. The strength of the magnetic field at the periphery of the loop is $B$ . Find the magnetic force on the wire.

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a

$B i (6 π a) \hat{k}$

b

$B i (2 π a) \hat{k}$

c

$B i (4 π a) \hat{k}$

d

$B i (π a) \hat{k}$

answer is A.

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Detailed Solution

Consider an element of length $d l$ , the force on it $d F = B i (d l) \sin 90^{°} = B i (d l)$ and perpendicular to $d \vec{l}$ and $\vec{B}$ that is along $z$ -axis. For any of the element of the loop, the direction is along z-axis. Therefore force on entire loop

$\vec{F} = \int_{0}^{2 π a} B i (d l) \hat{k} = B i \int_{0}^{2 π a} d l \hat{k} = B i (2 π a) \hat{k}$

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