Question

A magnetic flux of$\mathbf{7}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{mWb}$ is directed outward through the flat bottom face of the closed surface shown in Fig. 32-40. Along the flat top face (which has a radius of$\mathbf{4}\mathbf{.}\mathbf{2}\mathbf{}\mathbf{cm}$ ) there is a $\mathbf{0}\mathbf{.}\mathbf{40}\mathbf{}\mathbf{T}$magnetic field directed perpendicular to the face. What are the (a) magnitude and (b) direction (inward or outward) of the magnetic flux through the curved part of the surface?

Short answer

1. The magnitude of the magnetic flux is $9.2\times {10}^{-3}\mathrm{Wb}$.
2. The direction of the flux is inward.

Step-by-step solution

Listing the given quantities

Magnetic field is$0.40\mathrm{T}$

Radius of top face is

$\begin{array}{c}\mathrm{r}=4.2\mathrm{cm}\\ =4.2\times {10}^{-2}\mathrm{m}\end{array}$

Understanding the concepts of magnetic field

Here, we have to use the formula for the magnetic flux in terms of the magnetic field and area. As the total flux passing inward for any closed surface is equal to the total flux going out. we can find the flux through the curved surface.

Formula:

$\mathrm{A}={\mathrm{\pi r}}^2$

$\mathrm{\varphi }=\mathrm{BA}$

(a) Calculations of the magnitude of the magnetic flux.

The magnitude of the magnetic flux.

The area of the top face is as follow:

$\begin{array}{c}\mathrm{A}={\mathrm{\pi r}}^2\\ =\mathrm{\pi }{(4.2\times {10}^{-2})}^2\end{array}$

So, the flux passing through the top surface is

$\begin{array}{c}{\mathrm{\varphi }}_{\mathrm{top}}=\mathrm{BA}\\ =0.4\times \mathrm{\pi }\times {(4.2\times {10}^{-2})}^2\\ =2.125\times {10}^{-3}\mathrm{Wb}\end{array}$

So, the net flux inside the surface is

$\begin{array}{c}\mathrm{\varphi }={\mathrm{\varphi }}_{\mathrm{bottom}}+{\mathrm{\varphi }}_{\mathrm{top}}\\ =7.0\times {10}^{-3}+2.125\times {10}^{-3}\\ =9.21\times {10}^{-3}\mathrm{Wb}\end{array}$

Now, the total inward flux is equal to the total outward flux

${\mathrm{\varphi }}_{\mathrm{in}}={\mathrm{\varphi }}_{\mathrm{out}}$

Hence, the magnitude of the magnetic flux through the curved surface is

$\mathrm{\varphi }=9.21\times {10}^{-3}\mathrm{Wb}$

(b) Explanation of the direction of the flux

The direction of the flux.

As the flux is going out through both the top and the bottom flat surfaces, the direction of the flux through the curved surface must be inward.