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 sides a and,
(iii) a regular hexagon of sides a. The coil is connected to a voltage source `V_(0)` . Find the magnetic moment of the coils in each case.

Explanation

Magnetic dipole moment for this coil is m = nIA
(i) For equilateral triangle :
Side of equilateral triangle be a.
Total length of wire be 12 a.
`therefore` Number of turns of coil n = 3
Magnetic dipole moment for coil,
`m=nIA=4I(sqrt3/4a^(2))" "[becauseA=(sqrt3a^(2))/4]`
`m=Ia^(2)sqrt3`

(ii) For square coil :
Number of turn n = 3,
Area of square A = `a^(2)`
Magnetic dipole moment for square loop = nIA
= `3I(a^(2))`

(iii) For hexagon loop :
Number of turn n = 2.
Magnetic dipole moment,
m = nIA
= `2I((6sqrt3)/4a^(2))`
m = `3sqrt3Ia^(2)`