Chapter 33, Problem 1PQ

(a)

To determine

The inductor which has the greater inductance per unit length.

Answer to Problem 1PQ

The short and wide inductor has more inductance per unit length.

Explanation of Solution

Given that the two solenoids have the same number of turns per unit length, but one is short and wide and the other is long and narrow.

Write the expression for the inductance of a solenoid.

    $L={\mu }_0{n}^{2}Al$                                                    (I)

Here, $L$ is the inductance of a solenoid, ${\mu }_0$ is the permeability of free space, $n$ is the number of turns per unit length, $A$ is the cross section area of solenoid and $l$ is the length of the solenoid.

Rearrange the above equation to find inductance per unit length.

    $\frac{L}{l}={\mu }_0{n}^{2}A$                                                     (II)

Conclusion:

From the above equation (II), it is clear that inductance per unit length is directly proportional to the area of cross section of the solenoid for same number of turns per unit length.

    $\frac{L}{l}\propto A$

So, the inductance per unit length is more for the conductor having large cross section area.

Therefore, the short and wide inductor has more inductance per unit length.

(b)

To determine

The possibility for the two conductors to have the same inductance $L$.

Answer to Problem 1PQ

The inductance of two inductors will be same if the product of the cross sectional area and the length of the inductor is same for both the inductors.

Explanation of Solution

From equation (I), it is clear that if two coils have same number of turns per unit length, then the inductance of the coil is proportional to the product of the area of cross section and length of the coil.

    $L\propto Al$

So, the inductance of the two coils will be same if the product of area of cross section and length of the coil is same for both the coils.

Conclusion:

Therefore, the inductance of two inductors will be same if the product of the cross sectional area and the length of the inductor is same for both the inductors.