Chapter 19, Problem 90P

(a)

To determine

The $B_{\parallel }\Delta l$ for the four sides of the rectangle.

Answer to Problem 90P

The $B_{\parallel }\Delta l$ for the two perpendicular sides of the rectangular path and the side outside the solenoid is zero. The $B_{\parallel }\Delta l$ for the bottom side of the rectangular path is $Bb$.

Explanation of Solution

The path for the ampere’s law is rectangular. The magnetic field inside the solenoid is uniform and parallel to the axis of the solenoid. For the two sides of the rectangle which is perpendicular to the axis of the solenoid has magnetic field perpendicular to it for the portion it is inside the solenoid and for the portion outside the solenoid the magnetic field is zero. Thus $B_{\parallel }\Delta l$ is zero for the two perpendicular sides of the rectangle.

For the side which is outside the solenoid, the magnetic field is zero, thus $B_{\parallel }\Delta l$ is zero for the side of the rectangle outside the solenoid. For the bottom side of the rectangle which is inside solenoid, the magnetic field is parallel. The length of the bottom sides is $b$.

Write $B_{\parallel }\Delta l$ for the bottom side of the rectangle.

    $B_{\parallel }\Delta l=Bb$

Here, $B$ is the uniform magnetic field inside the solenoid, $b$ is the long side of the rectangular path, $B_{\parallel }$ is the parallel component of the magnetic field, $\Delta l$ is the length of the side.

Conclusion:

The $B_{\parallel }\Delta l$ for the two perpendicular sides of the rectangular path and the side outside the solenoid is zero. The $B_{\parallel }\Delta l$ for the bottom side of the rectangular path is $Bb$.

(b)

To determine

The circulation of the rectangular path.

Answer to Problem 90P

The circulation for the rectangular path is $Bb$.

Explanation of Solution

The circulation for the rectangular path is the sum of $B_{\parallel }\Delta l$ for all the sides of the rectangle.

The $B_{\parallel }\Delta l$ for the two perpendicular sides of the rectangular path and the side outside the solenoid is zero. The $B_{\parallel }\Delta l$ for the bottom side of the rectangular path is $Bb$.

Find the circulation for the rectangular path.

    $\begin{array}{c}{\displaystyle \sum B_{\parallel }\Delta l}=0+0+0+Bb\\ =Bb\end{array}$

Here, $B$ is the uniform magnetic field inside the solenoid, $b$ is the long side of the rectangular path, $B_{\parallel }$ is the parallel component of the magnetic field, $\Delta l$ is the length of the side.

Conclusion:

The circulation for the rectangular path is $Bb$.

(c)

To determine

To write the number of loops $N$ cut through the path in terms of turns per unit length $n$.

Answer to Problem 90P

The expression for the number of loops $N$ cut through the path in terms of turns per unit length $n$ is $N=nb$.

Explanation of Solution

Write the formula for the number of turns per unit length.

    $n=\frac{N}{L}$

Here, $n$ is the number of turns per unit length, $N$ is the number of loops cut through the path,

$L$ is the length.

The loops cut the path through the bottom side of the rectangular path.

    $n=\frac{N}{b}$

Here, $b$ is the length of the bottom side of the rectangle.

Conclusion:

The expression for the number of loops $N$ cut through the path in terms of turns per unit length $n$ is $N=nb$.

(d)

To determine

The magnetic field $B$.

Answer to Problem 90P

The magnetic field is ${\mu }_{0}nI$.

Explanation of Solution

The $B_{\parallel }\Delta l$ for the two perpendicular sides of the rectangular path and the side outside the solenoid is zero. The $B_{\parallel }\Delta l$ for the bottom side of the rectangular path is $Bb$.

Find the circulation for the rectangular path.

    $\begin{array}{c}{\displaystyle \sum B_{\parallel }\Delta l}=0+0+0+Bb\\ =Bb\end{array}$ (I)

Here, $B$ is the uniform magnetic field inside the solenoid, $b$ is the long side of the rectangular path, $B_{\parallel }$ is the parallel component of the magnetic field, $\Delta l$ is the length of the side.

From section (c)

  $b=\frac{N}{n}$

Here, $b$ is the length of the bottom side of the rectangular path, $N$ is the number of loops cut through the path, $n$ is the number of turns per unit length.

Substitute the above equation in equation (I).

    ${\displaystyle \sum B_{\parallel }\Delta l}=B\left(\frac{N}{n}\right)$                                                                                (II)

Write the formula for the circulation.

    ${\displaystyle \sum B_{\parallel }\Delta l}={\mu }_{0}NI$                                                                                  (III)

Here, ${\mu }_0$ is the permeability of the free space, $I$ is the current.

Compare equation (II) and (III).

    $\begin{array}{c}{\mu }_{0}NI=B\left(\frac{N}{n}\right)\\ B={\mu }_{0}nI\end{array}$

Conclusion:

The magnetic field is ${\mu }_{0}nI$.