The magnetic field $\overrightarrow{\mathbf{B}}$in a certain region is 0.128 T, and its direction is that of the +z-axis in below figure (a) What is the magnetic flux across the surface in the figure? (b) What is the magnetic flux across the surface aefd? (c) What is the magnetic flux across the surface befc? (d) What the net flux through all five surface that enclose the shaded volume.

The term magnetic flux may be defined as the measurement of total magnetic field passing through the given area.

The magnetic flux can be calculated as

From figure see that area vectors are perpendicular to the surface.

$$\begin{gathered} f_{abcd}=\overrightarrow{B}\times \overrightarrow{A} \\ {f}_{abcd}=BACos\left({90}^{\circ }\right) \\ {f}_{abcd}=0 \end{gathered}$$

Hence, the magnetic flux across the surface is$f_{abcd}=0\mathrm{Wb}$

The area of the surface in the opposite direction of magnetic field so the magnetic flux calculated as

$$\begin{gathered} f_{befo}=\overrightarrow{B}\times \overrightarrow{A} \\ {f}_{befo}=BACos\left({180}^{\circ }\right) \\ {f}_{beto}=-BA \\ {f}_{befc}=-B\left(be\right)\left(ef\right) \end{gathered}$$

Here

$$\begin{gathered} be=0.300m \\ ef=0.300m \end{gathered}$$

than

$$\begin{gathered} f_{beft}=-(0.128T)(0.300m)(0.300m) \\ {f}_{beft}=-0.0115Wb \end{gathered}$$

Hence, the magnetic flux across the surface is $f_{befc}=-0.0115\mathrm{Wb}$.

The angle between surface area and the magnetic field is

$$\begin{gathered} \cos \left(\theta \right)=\frac{30.0}{50.0} \\ \cos \left(\theta \right)=\frac{3}{5} \end{gathered}$$

Now the magnetic flux is

$$\begin{gathered} f_{aetd}=\overrightarrow{B}\times \overrightarrow{A} \\ {f}_{aefd}=\frac{3BA}{5} \\ {f}_{aefd}=\frac{3}{5}(0.128T)(0.500m)(0.300m) \\ {f}_{aefd}=+0.0115Wb \end{gathered}$$

Hence, the magnetic flux across the surface aefd is $f_{aefd}=+0.0115\mathrm{Wb}$.

And the net flux is zero because angle between them is$90°$ .than

$$\begin{gathered} f_{total}=f_{befic}+f_{aefd} \\ {f}_{tobl}=-0.0115Wb+0.0115Wb \\ {f}_{total}=0 \end{gathered}$$

Hence, the net flux through all five surface that enclose the shaded volume is 0Wb.