Question

There are two long co-axial solenoids of same length $l$. the inner and outer coils have radii $r_1$ and $r_2$ and number of turns per unit length $n_1$ and $n_2$ respectively. The rate of mutual inductance to the self-inductance of the inner-coil is :

• A. $\frac{n_{2}}{n_{1}}. \frac{r^{2}_{2}}{r_{1}^{2}} $
• B. $\frac{n_{2}}{n_{1}}. \frac{r_{1}}{r_{2}} $
• C. $\frac{n_1}{n_2}$
• D. $\frac{n_2}{n_1}$

Answer 1

The Correct Option is D

To find the ratio of mutual inductance to the self-inductance of the inner coil, we first need to understand the concepts of mutual inductance and self-inductance.

Mutual Inductance (M): When two solenoids are coaxial, the magnetic field created by an electrical current in one solenoid induces an electromotive force (emf) in the other solenoid. The mutual inductance is the proportionality constant that relates the induced emf in one solenoid to the rate of change of current in the other solenoid. For the given coaxial solenoids, the mutual inductance is given by:

M = \mu_0 n_1 n_2 \pi r_1^2 l

where \mu_0 is the permeability of free space, n_1 and n_2 are the number of turns per unit length for the inner and outer solenoids, respectively, and r_1 is the radius of the inner solenoid.

Self-Inductance (L) of the Inner Coil: This is the property of the coil itself to oppose the change in current passing through it. The self-inductance of a solenoid is given by:

L = \mu_0 n_1^2 \pi r_1^2 l

For the inner solenoid, this depends on the number of turns per unit length n_1, the length l, and the square of the radius r_1.

Ratio of Mutual Inductance to Self-Inductance:

To find the ratio \frac{M}{L}, we divide the expression for mutual inductance by the expression for self-inductance:

\frac{M}{L} = \frac{\mu_0 n_1 n_2 \pi r_1^2 l}{\mu_0 n_1^2 \pi r_1^2 l}

Simplifying this gives:

\frac{M}{L} = \frac{n_2}{n_1}

The above expression shows that the ratio of the mutual inductance to the self-inductance of the inner coil is simply the ratio of the number of turns per unit length of the outer solenoid to that of the inner solenoid.

Thus, the correct answer is:

\frac{n_2}{n_1}