Chapter 30, Problem 30.47P

A cube of edge length $\mathcal{l}=2.50$ cm is positioned as shown in Figure P30.47. A uniform magnetic field given by $\stackrel{\to }{B}$ = (5 $\hat{i}$ + 4 $\hat{j}$ + 3 $\hat{k}$) T exists throughout the region. (a) Calculate the magnetic flux through the shaded face. (b) What is the total flux through the six faces?

To determine

The magnetic flux through the shaded face.

Answer to Problem 30.47P

The magnetic flux through the shaded face is $3.13\text{mWb}$.

Explanation of Solution

Given info: The edge length $l$ of the cube is $2.50\text{cm}$ and uniform magnetic field is $\left(5\widehat{\text{i}}+4\widehat{\text{j}}+3\widehat{\text{k}}\right)\text{T}$.

Write the expression for the magnetic flux through the shaded face.

$\varphi =\overrightarrow{B}\cdot {\overrightarrow{A}}_{\text{x}}$ (1)

Here,

${\overrightarrow{A}}_{\text{x}}$ is the area of the square along the $x$ axis.

$\overrightarrow{B}$ is the magnetic field on the shaded portion.

Write the expression for the area of the square along the $x$ axis.

${\overrightarrow{A}}_{\text{x}}=l^2\widehat{\text{i}}$

Substitute $2.50\text{cm}$ for $l$ in the above expression for the area of square.

$\begin{array}{c}{\overrightarrow{A}}_{\text{x}}={\left(2.50\text{cm}\right)}^2\widehat{\text{i}}\\ ={\left(2.50\text{cm}\times \frac{1\text{m}}{100\text{cm}}\right)}^2\widehat{\text{i}}\\ =6.25\times {10}^{-4}\widehat{\text{i}}{\text{m}}^{\text{2}}\end{array}$

Thus, the area of the square is $6.25\times {10}^{-4}\widehat{\text{i}}{\text{m}}^{\text{2}}$.

Substitute $\left(5\widehat{\text{i}}+4\widehat{\text{j}}+3\widehat{\text{k}}\right)\text{T}$ for $\overrightarrow{B}$ and $6.25\times {10}^{-4}\widehat{\text{i}}{\text{m}}^{\text{2}}$ for ${\overrightarrow{A}}_{\text{x}}$ in the equation (1) for the magnetic flux through the shaded face.

$\begin{array}{c}\varphi =\left(5\widehat{\text{i}}+4\widehat{\text{j}}+3\widehat{\text{k}}\right)\text{T}\cdot 6.25\times {10}^{-4}\widehat{\text{i}}{\text{m}}^{\text{2}}\\ =3.125\times {10}^{-3}{\text{T-m}}^{\text{2}}\times \frac{1\text{Wb}}{1{\text{T-m}}^{\text{2}}}\\ =3.125\times {10}^{-3}\text{Wb}\times \frac{1000\text{m-Wb}}{1\text{Wb}}\\ =3.13\text{m-Wb}\end{array}$

Conclusion:

Therefore, the magnetic flux through the shaded face is $3.13\text{mWb}$.

To determine

The total flux through the six faces.

Answer to Problem 30.47P

The total flux through the six faces is $0$.

Explanation of Solution

Given info: The edge length $l$ of the cube is $2.50\text{cm}$ and uniform magnetic field is $\left(5\widehat{\text{i}}+4\widehat{\text{j}}+3\widehat{\text{k}}\right)\text{T}$.

Write the expression for the magnetic flux through the six faces.

${\varphi }_{\text{T}}={\varphi }_1+{\varphi }_2+{\varphi }_3+{\varphi }_4+{\varphi }_5+{\varphi }_6$ (1)

Here,

${\varphi }_1$ is the flux passes through the face $1$.

${\varphi }_2$ is the flux passes through the face $2$.

${\varphi }_3$ is the flux passes through the face $3$.

${\varphi }_4$ is the flux passes through the face $4$.

${\varphi }_5$ is the flux passes through the face $5$.

${\varphi }_6$ is the flux passes through the face $6$.

Write the expression for the flux passes through the face $1$.

${\varphi }_1=\overrightarrow{B}\cdot {\overrightarrow{A}}_{\text{x}}$

Here,

${\overrightarrow{A}}_{\text{x}}$ is the area of the square along the $x$ axis.

$\overrightarrow{B}$ is the magnetic field on the face $1$.

Write the expression for the flux passes through the face $1$.

${\varphi }_2=-\overrightarrow{B}\cdot {\overrightarrow{A}}_{\text{x}}$

Here,

${\overrightarrow{A}}_{\text{x}}$ is the area of the square along the negative $x$ axis.

$\overrightarrow{B}$ is the magnetic field on the face $2$.

Write the expression for the flux passes through the face $3$.

${\varphi }_3=\overrightarrow{B}\cdot {\overrightarrow{A}}_{\text{y}}$

Here,

${\overrightarrow{A}}_{\text{y}}$ is the area of the square along the $y$ axis.

$\overrightarrow{B}$ is the magnetic field on the face $3$.

Write the expression for the flux passes through the face $4$.

${\varphi }_4=-\overrightarrow{B}\cdot {\overrightarrow{A}}_{\text{y}}$

Here,

${\overrightarrow{A}}_{\text{y}}$ is the area of the square along the negative $y$ axis.

$\overrightarrow{B}$ is the magnetic field on the face $4$.

Write the expression for the flux passes through the face $5$.

${\varphi }_5=\overrightarrow{B}\cdot {\overrightarrow{A}}_{\text{z}}$

Here,

${\overrightarrow{A}}_{\text{z}}$ is the area of the square along the $z$ axis.

$\overrightarrow{B}$ is the magnetic field on the face $5$.

Write the expression for the flux passes through the face $6$.

${\varphi }_6=-\overrightarrow{B}\cdot {\overrightarrow{A}}_{\text{z}}$

Here,

${\overrightarrow{A}}_{\text{z}}$ is the area of the square along the negative $z$ axis.

$\overrightarrow{B}$ is the magnetic field on the face $6$.

Substitute $\overrightarrow{B}\cdot {\overrightarrow{A}}_{\text{x}}$ for ${\varphi }_1$, $-\overrightarrow{B}\cdot {\overrightarrow{A}}_{\text{x}}$ for ${\varphi }_2$, $\overrightarrow{B}\cdot {\overrightarrow{A}}_{\text{y}}$ for ${\varphi }_3$, $-\overrightarrow{B}\cdot {\overrightarrow{A}}_{\text{y}}$ for ${\varphi }_4$, $\overrightarrow{B}\cdot {\overrightarrow{A}}_{\text{z}}$ for ${\varphi }_5$ and $-\overrightarrow{B}\cdot {\overrightarrow{A}}_{\text{z}}$ for ${\varphi }_6$ in the above expression for the total magnetic flux passes through the six faces of the cube.

$\begin{array}{c}{\varphi }_{\text{T}}=\overrightarrow{B}\cdot {\overrightarrow{A}}_{\text{x}}-\overrightarrow{B}\cdot {\overrightarrow{A}}_{\text{x}}+\overrightarrow{B}\cdot {\overrightarrow{A}}_{\text{y}}-\overrightarrow{B}\cdot {\overrightarrow{A}}_{\text{y}}+\overrightarrow{B}\cdot {\overrightarrow{A}}_{\text{z}}-\overrightarrow{B}\cdot {\overrightarrow{A}}_{\text{z}}\\ =0\end{array}$

Conclusion:

Therefore, the total flux through the six faces of the square cube is $0$.