Question

A loop, made of straight edges has six corners at $A(0,0,0),B(L,0,0)C(L,L,0),D(0,L,0),E(0,L,L)and\left(F\right(0,0,L).$
A magnetic field $\overrightarrow{B}=B_{\circ }(\hat{i}+\hat{k})$ T is present in the region. The flux passing through the loop $ABCDEFA$ (in that order) is:

• A. $B_{\circ }{L}^2$ Wb
• B. $4B_{\circ }{L}^2$ Wb
• C. $\sqrt{2}{B}_{\circ }{L}^2$ Wb
• D. $2B_{\circ }{L}^2$ Wb

Answer 1

The correct option is D

$2B_{\circ }{L}^2$ Wb

Given, $\overrightarrow{B}=B_{\circ }(\hat{i}+\hat{k})$ T

Area, $\overrightarrow{A_1}=L^2\hat{k}$
$\overrightarrow{A_2}=L^2\hat{i}$

Magnitude of flux passing through the loop $ABCDEFA,$

$\varphi ={\varphi }_1+{\varphi }_2$

$\varphi =\overrightarrow{B}.{\overrightarrow{A}}_1+\overrightarrow{B}.\overrightarrow{A_2}$

$=\left\{B_{\circ }\right(\hat{i}+\hat{k}\left)\right\}.\left(L^2\hat{k}\right)+\left\{B_{\circ }\right(\hat{i}+\hat{k}\left)\right\}.\left(L^2\hat{i}\right)$

$=\left(2B_{\circ }{L}^2\right)$ Wb

Final Answer: Option (B)