Question

The magnetic field in a region of space varies according to the equation$\overrightarrow{B}=B_0[x\hat{i}+y\hat{j}+\hat{k}]$. The magnetic flux associated with the area bounded by the line $x=0,y=0,x=a$ and $y=\frac{b}{2}$ in the $x-y$ plane is

• A. $\text{zero}$
• B. $B_{0}ba$
• C. $\frac{3B_{0}ba}{2}$
• D. $\frac{B_{0}ba}{2}$

Answer 1

The correct option is D $\frac{B_{0}ba}{2}$


Magnetic flux ${\varphi }_B$ is given by

${\varphi }_B=\overrightarrow{B}\cdot \overrightarrow{A}$

From the diagram, we can see that the area vector is along $z-$ axis,

$\overrightarrow{A}=A\hat{k}=\frac{ba}{2}\hat{k}$

and, $\overrightarrow{B}=B_0[x\hat{i}+y\hat{i}+\hat{k}]$

So, the magnetic flux lying in the $xy-$plane i.e., for, $x=0,y=0$

$\overrightarrow{B}=B_0[0\hat{i}+0\hat{j}+\hat{k}]$

$\Rightarrow {\varphi }_B=\overrightarrow{B}\cdot \overrightarrow{A}$

$=B_0\left[\hat{k}\right]\cdot \frac{ba}{2}\hat{k}=\frac{B_{0}ba}{2}$

Hence, option $\left(D\right)$ is correct.