An iron ring of relative permeability µr has windings of insulated copper wire of n turns per meter. When the current in the windings is I, find the expression for the magnetic field in the ring.
An iron ring of relative permeability µr having windings of insulated copper wire of n turns per meter can be considered as a toroid. Consider a toroidal solenoid with centre O as shown.
r = average radius of the toroid
I = current through the solenoid
To determine the magnetic field produced at the centre along the axis of the toroid due to current I, we imagine an Amperian loop of radius r and traverse it in the clockwise direction.
According to Ampere's circuital law, we have:
`ointvecB.vec(dl) = mu_rI`
The total current flowing through the toroid is NI, where N is the total number of turns.
`ointvecB.vec(dl) = mu_r(NI)` ......(1)
Now `vecB` and `vec(dl)` are in the same direction
`oint vecB.vec(dl) = B.ointdl`
`=> ointvecb.vec(dl) = B(2pir)` ....(2)
Comparing equations (1) and (2), we get:
`B(2pir) = mu_rNI`
`=> B = (mu_rNI)/(2pir)`
Now it is given that n is the number of turns per unit length, then
`n = N/(2pir)`
∴`B = mu_0nI` which is the expression for the magnetic field in the ring.