Question

Magnetic field in a region is given by $\overrightarrow{B}=B_{o}x\hat{k}$. Two loops each of side a is placed in this magnetic region in the x-y plane with one of its sides on x-axis. If $F_1$ is the force on loop 1 and $F_2$ be the force on loop 2 then:

• A. $F_1=F_2=0$
• B. $F_1>${F}_{2}$
• C. $F_2>F_1$
• D. $F_1=F_2\ne 0$

Answer 1

Answer: (D)

$F_1=F_2\ne 0$

Forces on the wires parallel to X-axis will be obtained by integration (as $B\propto X$ and x coordinates vary along these wires). But on a loop, there are two such wires. Force on them will be equal and opposite.
Forces on two wires parallel to y-axis can be obtained directly (without integration) as the value of B is same along these wires. But their values will be different (as x-coordinate and therefore B is different).
$F_{net}=\Delta F$ ( on two wires)
$=Ia\left(\Delta B\right)=Ia\left(B_o\right)\left(\Delta x\right)$
$=Ia{B}_o\left(a\right)=I{B}_o{a}^2$
This is independent of x
$\therefore F_1=F_2=IBl{a}^2\ne 0$