A long insulated copper wire is closely wound as a spiral of `N` turns. The spiral has inner radius a and outer radius `b`. The spiral lies in the `xy`-plane and a steady current I flows through the wire. The`z`-component of the magetic field at the centre of the spiral is

Options

A. `(mu_(0)NI)/(b)`
B. `(2mu_(0)NI)/(a)`
C. `(mu_(0)NI)/(2 (b-a)) ln "(b)/(a)`
D. `(mu_(0)I^(N))/(2(b-a))ln " (b)/(a)`

Answer

`C`

Explanation

`dN = (N)/((b -a)) dx`
`dB = (mu_(0))/(4pi) (2pi (dN)I)/(x) = (mu_(0))/(4pi) (2piNdx)/((b-a)) (I)/(x)`
`B = (mu_(0))/(2) (NI)/((b-a)) underset(a)overset(b)int (dx)/(x) = (mu_(0))/(2) (NI)/((b-a)) log_(e)" (b)/(a)`
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