Question

A wire lying along y-axis from y=0 to y=1 m carries a current of 2 mA in the negative y-direction.The wire lies in a non-uniform magnetic field given by $\overrightarrow{B}=\left(0.3T/m\right)y\hat{i}+\left(0.4Tm\right)y\hat{j}$.The magnetic force on the entire wire is

• A. $-3\times {10}^{-4}\hat{j}N$
• B. $6\times {10}^{-3}\hat{k}N$
• C. $-3\times {10}^{-4}\hat{k}N$
• D. $3\times {10}^{-4}\hat{k}N$

Answer 1

Answer: (D)

$3\times {10}^{-4}\hat{k}N$

Due to $\hat{j}$ component of $\overrightarrow{B}$, magnetic force is zero.It is only due to $\hat{i}$ component.
$\overrightarrow{l}$ is towards -y direction $\overrightarrow{B}$ is towards $\hat{i}$ direction.
Hence $\overrightarrow{l}\times \overrightarrow{B}$ (the direction of magnetic force is +ve -Z)

$F=\int dF={\int }_0^1\left(i\right)\left(dy\right)\left(B\right)$

$={\int }_0^1(2\times {10}^{-3})dy\left(0.3y\right)$

$=3\times {10}^{-4}N$