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

We know that magnetic moment of the coils m=nlA.
Since, the same wire is used in three cases with same potential, therefore, same current flows in three cases.
(i) for an equilateral triangle of side a,
`n=4` as the total wire of length =12a
Magnetic moment of the coils `m=nIA=4I((sqrt(3))/(3)a^(2))`
`thereforem=Ia^(2)sqrt(3)`.
(ii). For a square of sides a,
n=3 as the total wire of length=12a
Magnetic moemnt of the coils m=nIA=3I`(a^(2))=3Ia^(2)`
(iii). For a regular hexagon of sides a,
n=2 as the total wire of length =12a
Magnetic moment of the coils `m=nlA=21((6sqrt(3))/(4)a^(2))`
`m=3sqrt(3)a^(2)I`
m is a geometric series.