Question

Two solenoids of equal number of turns have their lengths and the radii in the same ratio 1 : 2. The ratio of their self inductances will be

• A. 1 :2
• B. 2 : 1
• C. 1 : 1
• D. 1 :4

Answer 1

Answer: (A)

1 :2

Answer 2

Self inductance of a solenoid,

$L = \frac{\mu_{0}N^{2}A}{l} = \frac{\mu_{0}N^{2}\pi r^{2}}{l}$

Where l is the length of the solenoid, N is the total number of turns of the solenoid and A is the area of cross section of the solenoid.

$\therefore\frac{L_{1}}{L_{2}} = \left( \frac{N_{1}}{N_{2}} \right)^{2}\left( \frac{r_{1}}{r_{2}} \right)^{2}\left( \frac{l_{2}}{l_{1}} \right)$

Here $N_{1} = N_{2},\frac{l_{1}}{l_{2}} = \frac{1}{2},\frac{r_{1}}{r_{2}} = \frac{1}{2}$

$\therefore\frac{L_{1}}{L_{2}} = \left( \frac{1}{2} \right)^{2}\left( \frac{2}{1} \right) = \frac{1}{2}$