Question

A uniform conducting wire of length 12a and resistance ‘R’ is wound up as a current-carrying coil in the shape of, 
(i) an equilateral triangle of side ‘a’. 
(ii) a square of side ‘a’. 
The magnetic dipole moments of the coil in each case respectively are:

• A. 4Ia² and 3Ia²
• B. √3Ia² and 3Ia²
• C. 3Ia² and Ia²
• D. 3Ia² and 4Ia²

Answer 1

The Correct Option is B

To determine the magnetic dipole moments of the coil when the wire is wound into different shapes, we need to apply the formula for the magnetic dipole moment \( M \) of a current loop:

\(M = I \times A\)

where \(I\) is the current flowing through the loop, and \(A\) is the area enclosed by the loop.

1. Equilateral Triangle of side \( a \):
   • The perimeter of the triangle is \(3a\).
   • The wire is of length \(12a\), which can form four loops of the equilateral triangle.
   • Area of one equilateral triangle is given by:
   • The total magnetic dipole moment for 4 such loops is:
2. Square of side \( a \):
   • The perimeter of the square is \(4a\).
   • The wire is of length \(12a\), which can form three loops of the square.
   • Area of one square is:
   • The total magnetic dipole moment for 3 such loops is:

Thus, the magnetic dipole moments for the equilateral triangle and square formations are \(\sqrt{3} Ia^2\) and \(3 Ia^2\), respectively.

The correct answer is therefore:

\(\sqrt{3} Ia^2\) and \(3 Ia^2\).