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Q)

A loop, made of straight edges has six corner at A(0, 0, 0), B(L, 0, 0), C(L, L, O), D(O, L,O), E(O, L, L) and F (0, 0, L). A magnetic field $B = B_o(\hat{i} + \hat{k})T$ is present in the region.
The flux passing through the loop ABCDEFA (in that order) is $(a)B_oL^2Wb$ $(b)2B_oL^2Wb$ $(c)\sqrt 2B_oL^2Wb$ $(d)4B_oL^2Wb$



$(a)B_oL^2Wb$
$(b)2B_oL^2Wb$
$(c)\sqrt 2B_oL^2Wb$
$(d)4B_oL^2Wb$

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A)
Given :
$\overrightarrow {B} = B_o(\hat{i} + \hat{k})$
Area vector of $ ABCD = L^2 \hat{k}$
Area vector of $DEFA = L^2\hat{i}$
Total area vector, $\overrightarrow{A} =L^2(\hat{i} + \hat {k})$
$\begin{align*} \text{Total magnetic flux } \; \phi & = \overrightarrow {B}. \overrightarrow{A} \\ & = B_o (\hat{i} + \hat{k}).L^2(\hat{i} + \hat{k}) \\ & = B_oL^2(1+1) \\ & = 2 B_o L^2 wb \end{align*}$
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