Question

A uniform conducting wire of length is 24a, and resistance R is wound up as a current carrying coil in the shape of an equilateral triangle of side 'a' and then in the form of a square of side 'a'. The coil is connected to a voltage source V\(_0\). The ratio of magnetic moment of the coils in case of equilateral triangle to that for square is 1 : \(\sqrt{y}\) where y is _________.

Hint:

For a given length of wire turned into a multi-turn coil, the magnetic moment is \(M = N I A = (\frac{L}{P}) I A\), where P is the perimeter. This shows that for a fixed side length, the number of turns plays a crucial role. Always calculate N first.

Answer 1

Correct Answer: 3

Let's analyze the problem: we need to find the ratio of the magnetic moments of a coil formed into the shape of an equilateral triangle and a square using the same length of wire. The magnetic moment \(M\) is given by \(M = nIA \), where \(n\) is the number of turns, \(I\) is the current, and \(A\) is the area of the loop.
Given, the total length of wire is 24a, and resistance is \(R\).
Step 1: Equilateral Triangle
The wire length forming an equilateral triangle is \(3a\) per turn. Hence, the number of turns \(n_t = \frac{24a}{3a} = 8\).
The area \(A_t\) of an equilateral triangle of side \(a\) is \(\frac{\sqrt{3}}{4}a^2\).
Step 2: Square
The wire length forming a square is \(4a\) per turn. Hence, the number of turns \(n_s = \frac{24a}{4a} = 6\).
The area \(A_s\) of a square of side \(a\) is \(a^2\).
Step 3: Current Through the Coils
Since resistance \(R = \frac{24a}{L}R_0\), for 1 turn, \(R_t = \frac{R}{8}\) and \(R_s = \frac{R}{6}\).
The current for both configurations \(I = \frac{V_0}{R}\) remains constant due to the same applied voltage and total wire resistance.
Step 4: Calculate Magnetic Moments
For the triangle, \(M_t = n_tIA_t = 8I\frac{\sqrt{3}}{4}a^2 = 2\sqrt{3}Ia^2\).
For the square, \(M_s = n_sIA_s = 6Ia^2\).
Step 5: Calculate the Ratio
\(\text{Ratio} = \frac{M_t}{M_s} = \frac{2\sqrt{3}Ia^2}{6Ia^2} = \frac{\sqrt{3}}{3}\). This ratio is \(1:\sqrt{y}\) where \(\sqrt{y} = \frac{\sqrt{3}}{3}\), hence \(y = 3\).
Conclusion
The value of \(y\) is 3, which falls within the expected range of 3 to 3.