A square of side L m lies in XY plane in a region, where the magnetic field is given by \( \vec{B} = B_0(2\hat{\imath}+3\hat{\jmath}+4\hat{k})\,\mathrm{T} \) and \( B_0 \) a constant. The magnitude of the flux passing through the square is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( 4B_0L^{2}\,\mathrm{Wb} \)
The magnetic flux through a flat area is \( \Phi = \vec{B}\cdot\vec{A} \). The square lies in the XY plane, so its area vector is perpendicular to the plane, along \( \hat{k} \): \( \vec{A} = L^{2}\hat{k} \).
\( \Phi = B_0(2\hat{\imath}+3\hat{\jmath}+4\hat{k})\cdot L^{2}\hat{k} = 4B_0L^{2}\,\mathrm{Wb} \)
Only the z-component of the field, \( 4B_0 \), crosses the plane; the x and y components run along the plane and contribute nothing.