Chapter 20, Problem 30Q

(a)

To determine

To explain: The effect on magnetic field inside a long solenoid for the given changes.

Explanation of Solution

Given:

The magnetic field inside a long solenoid is B .

The diameter of all the loops is doubled.

Formula used:

The magnetic field B inside a solenoid is given by,

  $B=\frac{{\mu }_{o}NI}{L}$

Where ${\mu }_o$ is the permeability of free space, $N$ is the number of loops of the solenoid, $I$ is the current flowing through the solenoid, and $L$ is the length of the solenoid.

The magnetic field B inside a solenoid does not depend on the diameter of the loops of the solenoid. So any change in the diameter of the loops does not affect the magnetic field inside the solenoid.

Conclusion:

If the diameter of all the loops is doubled then the magnetic field inside the solenoid remains unchanged.

(b)

To determine

To explain: The effect on magnetic field inside a long solenoid for the given changes.

Explanation of Solution

Given:

The magnetic field inside a long solenoid is B .

The spacing between the loops is doubled.

Formula used:

  $B=\frac{{\mu }_{o}NI}{L}$

Calculation:

The magnetic field inside a solenoid depends on the length of the solenoid. If the spacing between the loops of the solenoid is increased then the length of the solenoid increases and hence the magnetic field decreases. If the spacing between the loops gets doubled then the length of the solenoid gets doubled. Let the new length and the new magnetic field be $L\text{'}$ and $B\text{'}$ , respectively.

  $\begin{array}{l}\because L\text{'}=2L\\ \therefore B\text{'}=\frac{{\mu }_{o}NI}{L\text{'}}=\frac{{\mu }_{o}NI}{2L}=\frac{B}{2}\end{array}$

Conclusion:

If the spacing between the loops of a solenoid gets doubled, the magnetic field inside the solenoid gets reduced to half.

(c)

To determine

To explain: The effect on magnetic field inside a long solenoid for the given changes.

Explanation of Solution

Given:

The magnetic field inside a long solenoid is B .

The length of solenoid and number of loops are doubled.

Formula used:

  $B=\frac{{\mu }_{o}NI}{L}$

Calculation:

The magnetic field inside a solenoid is inversely proportional to the length of the solenoid and proportional to the number of loops, so any change in either the number of loops or the length results in a change in the magnetic field.

Let $N\text{'}$ and $L\text{'}$ be the new number of loops and the new length of the solenoid, respectively. The number of loops and the length of the solenoid are doubled then,

  $N\text{'}=2N\text{ and }L\text{'}=2L$

Let $B\text{'}$ be the new magnetic field then,

  $B\text{'}=\frac{{\mu }_{o}N\text{'}I}{L\text{'}}=\frac{{\mu }_{o}2NI}{2L}=B$

Conclusion:

If both the length as well as the number of loops of a solenoid gets doubled, the magnetic field inside the solenoid does not change.