Chapter 30, Problem 1OQ

(i)

To determine

The magnitude of the magnetic field inside a long solenoid, if the current is doubled.

Answer to Problem 1OQ

Option (b) It becomes twice as large; is correct.

Explanation of Solution

Write the expression for magnetic field inside a solenoid.

    $B={\mu }_{0}nI$                                                                                                                  (I)

Here, ${\mu }_0$ is the permeability of free space, $n$ is the turn density, $I$ is the current and $B$ is the magnitude of the magnetic field.

Substitute $2I$ for $I$ in equation (I).

    $\begin{array}{c}B={\mu }_{0}n\left(2I\right)\\ =2{\mu }_{0}nI\end{array}$

If the current inside a long solenoid is doubled, the magnitude of the magnetic field becomes twice as large.

Conclusion:

Therefore, option (b) is correct.

(ii)

To determine

The magnitude of the magnetic field inside a long solenoid, if the length of the solenoid is doubled.

Answer to Problem 1OQ

Option (d) It becomes one-half as large; is correct.

Explanation of Solution

Substitute $\frac{N}{l}$ for $n$ in equation (I) to find $B$.

    $B={\mu }_0\left(\frac{N}{l}\right)I$                                                                    (II)

Here, $N$ is the number of turns and $l$ is the length of the solenoid.

Substitute $2l$ for $l$ in equation (II) to find $B$.

    $\begin{array}{c}B={\mu }_0\left(\frac{N}{2l}\right)I\\ =\frac{1}{2}{\mu }_0\left(\frac{N}{l}\right)I\end{array}$

If the length of the solenoid is doubled, then the magnitude of the magnetic field becomes one-half as large.

Conclusion:

Therefore, option (d) is correct.

(iii)

To determine

The magnitude of the magnetic field inside a long solenoid, if the number of turns is doubled.

Answer to Problem 1OQ

Option (b) It becomes twice as large; is correct.

Explanation of Solution

Substitute $2N$ for $N$ in equation (II) to find $B$.

    $\begin{array}{c}B={\mu }_0\left(\frac{2N}{l}\right)I\\ =2\left(\frac{{\mu }_{0}NI}{l}\right)\end{array}$

If the number of turns of the solenoid is doubled, then the magnitude of the magnetic field becomes twice as large.

Conclusion:

Therefore, option (b) is correct.

(iv)

To determine

The magnitude of the magnetic field inside a long solenoid, if the radius doubled.

Answer to Problem 1OQ

Option (c) It is unchanged; is correct.

Explanation of Solution

Equation (I) signifies that the magnitude of the magnetic field depends upon the current, permeability of free space and the turn density. It does not depend upon the radius.

Therefore, the magnitude of the magnetic field remains unchanged if the radius is doubled.

Conclusion:

Therefore, option (c) is correct.