A square loop of edge length 2 m carrying a current of 2 A is placed with its edges parallel to the x-y axis. A magnetic field is passing through the x-y plane and expressed as where . The net magnetic force experienced by the loop is ______ N.

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Solution

Q.
A square loop of edge length 2 m carrying a current of 2 A is placed with its edges parallel to the x-y axis. A magnetic field is passing through the x-y plane and expressed as $\vec{B} = B_{0} (1 + 4 x) \hat{k},$ where $B_{0} = 5 T .$ . The net magnetic force experienced by the loop is ______ N. [2024]

Ans.
(160)

$B (x = 0) = B_{0}, B (x = 2) = 9 B_{0}$

Also, $\vec{F} = i \vec{ℓ} \times \vec{B}$

${\vec{F}}_{1} = i ℓ B_{0} (\hat{j} \times \hat{k}) = i ℓ B_{0} (\hat{i}) = 2 \times 2 \times 5 (\hat{i}) = 20 N (\hat{i})$

${\vec{F}}_{2} = 9 i ℓ B_{0} (- \hat{j} \times \hat{k}) = 9 i ℓ B_{0} (- \hat{i}) = 2 \times 2 \times 45 (- \hat{i}) = 180 N (- \hat{i})$

$F_{net} on AD and CB = 0$

${\vec{F}}_{net} = {\vec{F}}_{2} - {\vec{F}}_{1} = 160 N (- \hat{i})$

$\Rightarrow F_{net} = 160 N$

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