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Physics-

General

Easy

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

1:2
2:1
1:1
1:4

The correct answer is: 1:2

Self inductance of a solenoid
$L equals fraction numerator mu subscript 0 end subscript N to the power of 2 end exponent A over denominator l end fraction equals fraction numerator mu subscript 0 end subscript N to the power of 2 end exponent pi r to the power of 2 end exponent over denominator l end fraction$
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
$therefore fraction numerator L subscript 1 end subscript over denominator L subscript 2 end subscript end fraction equals open parentheses fraction numerator N subscript 1 end subscript over denominator N subscript 2 end subscript end fraction close parentheses to the power of 2 end exponent open parentheses fraction numerator r subscript 1 end subscript over denominator r subscript 2 end subscript end fraction close parentheses to the power of 2 end exponent open parentheses fraction numerator l subscript 2 end subscript over denominator l subscript 1 end subscript end fraction close parentheses$
Here, $N subscript 1 end subscript equals N subscript 2 end subscript comma fraction numerator l subscript 1 end subscript over denominator l subscript 2 end subscript end fraction equals fraction numerator 1 over denominator 2 end fraction comma fraction numerator r subscript 1 end subscript over denominator r subscript 2 end subscript end fraction equals fraction numerator 1 over denominator 2 end fraction$
$therefore fraction numerator L subscript 1 end subscript over denominator L subscript 2 end subscript end fraction equals open parentheses fraction numerator 1 over denominator 2 end fraction close parentheses to the power of 2 end exponent open parentheses fraction numerator 2 over denominator 1 end fraction close parentheses equals fraction numerator 1 over denominator 2 end fraction$

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