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 $F(0, 0, L)$. A magnetic field $\vec{B} = B_0 (\hat{i} +\hat{ k}) T$ is present in the region. The flux passing through the loop $A B C D E F A$ (in that order) is

• A. $B_0L^2Wb$
• B. $2B_0L^2Wb$
• C. $\sqrt{2}B_0 L^2Wb$
• D. $4B_0L^2Wb$

Answer 1

Answer: (B)

Here, $\vec{B} = B_0 (\hat{i} + \hat{k}) T$

Area vector of $ABCD = L^2 \hat{k}$
Area vector of $DEFA = L^2 \hat{i}$
Total area vector, $\vec{A} = L^2 (\hat{i} + \hat{k})$
Total magnetic flux, $\phi = \vec{B} \cdot \vec{ A}$
$= B_{0} \left(\hat{i} +\hat{k}\right)\cdot L^{2}\left(\hat{i}+\hat{k}\right) $
$= B_{0}L^{2} \left(1+1\right)$
$= 2 B_{0} L^{2} Wb$