For a given length l of wire carrying a current l, how many circular turns would produce the maximum magnetic moment ?

Explanation

For a cicular coil having N turns, magnetic moment
`M=piR^(2)iN...(i)`
Now, length of wire `L=(2piR)N`
`therefore R=(L)/(2piN)`
Substituting the above value of R in equation (i), we get
`M=piNixx(L^(2))/(4pi^(2)N^(2))or,M=(iL^(2))/(4piN)`
From equation (ii), it is clear that M will be maximum when N = minimum = 1, i.e., the coil has only one turn and
`(M)_(max)=(1)/(4pi)iL^(2)`