Electromagnetic Induction and Alternating Current 3 Question 4
####4. A copper wire is wound on a wooden frame, whose shape is that of an equilateral triangle. If the linear dimension of each side of the frame is increased by a factor of 3 , keeping the number of turns of the coil per unit length of the frame the same, then the self-inductance of the coil
(Main 2019, 11 Jan II)
(a) increases by a factor of 3 .
(b) decreases by a factor of $9 \sqrt{3}$.
(c) increases by a factor of 27 .
(d) decreases by a factor of 9 .
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Answer:
Correct Answer: 4. (c)
Solution:
- Self-inductance of a coil is given by the relation
$$ L=\mu_{0} n^{2} A \cdot l $$
where, $n$ is number of turns per unit length. Shape of the wooden frame is equilateral triangle.
$\therefore$ Area of equilateral triangle,
$$ A=\frac{\sqrt{3}}{4} a^{2} $$
(where, $a$ is side of equilateral triangle)
$\therefore$ Self-inductance, $L=\mu_{0} n^{2} \frac{\sqrt{3}}{4} a^{2} \cdot l$
Here, $\quad l=3 a \times N$ (where, $N$ is total number of turns) $\therefore \quad L=\mu_{0} n^{2} \frac{\sqrt{3}}{4} a^{2} \times 3 a N$ or $L \propto a^{3}$
When each side of frame is increased by a factor 3 keeping the number of turns per unit length of the frame constant.
Then,
$$ a^{\prime}=3 a $$
$$ \begin{array}{ll} \therefore & L^{\prime} \propto\left(a^{\prime}\right)^{3} \text { or } L^{\prime} \propto(3 a)^{3} \\ \text { or } & L^{\prime} \propto 27 a^{3} \text { or } L^{\prime}=27 L \end{array} $$
