A current carrying circular loop of radius R is placed in the x-y plane with centre at the origin. Half of the loop with `xgt0` is now bent so that it now lies in the y-z plane.

Options

A. The magnitude of magnetic moment now diminishes.
B. The magnetic moment does not change.
C. The magnitude of B at `(0.0 .z) , z gt gt R` increases.
D. The magnitude of B at `(0.0.z) ,z gt gt R` is unchanged.

Answer

a

Explanation

Let the loop be carrying current is an anticlockwise direction.
The area vector of the loop will be along positive Z-direction. For a circular loop of radius R, carrying current I, themagnetic moment will be given by:

`M =I.A = I(pi R^(2))` [Along positive Z - axis]
Now, half of the loop is bent in the Y-Z plane (part ACD).
So, the net magnetic moment of the loop will be equal to the sum of the two magnetic moments of parts ABD and ACD of the loop.
Magnetic moment of part ABD will be
` M_(1) = I.A_(1) = I (pi R^(2))//2`
[Along positive Z - axis]
Magnetic moment of part ACD will be
` M_(2) = I.A_(2) = I (piR^(2))//2`
[Along positive X - axis]
` :. ` Resultant magnetic moment will be
`M. = sqrt((M_(1))^(2) + (M_(2))^(2)) = sqrt(((IpiR^(2))/2)^(2) + ((IpiR^(2))/2)^(2))`
` = (I piR^(2))/2 sqrt2 ` [ Along Z-X plane at an angle `45^(@)` with X -axis]
` :. M. lt M`
Thus, the magnitude of magnetic moment diminishes.
The magnitude of `vec B` at a point on the axis of loop at a distance from the centre of the loop lying in the X-Y plane is:
` B = mu_(0)/(4 pi) (2 pi IR^(2))/((R^(2) + z^(2))^(3//2))` [towards positive Z - axis] ....(i)
The magnitude of `vec B` at a point due to part ABD will be
`B_(1) = B/2 = mu_(0)/(4 pi) (pi I R^(2))/((R^(2) + z^(2))^(3//2))` [towards positive Z - axis ]
`:. ` The resultant magnetic field at a point on the axis of loop at a distance Z from the centre will be:
`B.. = sqrt(B_(1)^(2) + B_(2)^(2))`
` = sqrt((mu_(0)/(4 pi) (piIR^(2))/((R^(2)+ z^(2))^(3//2)))^(2) + (mu_(0)/(4 pi) (piIR^(2))/((R^(2) + z^(2))^(3//2))))`
` = mu_(0)/(4 pi) (pi IR^(2))/((R^(2) + z^(2))^(3//2)) sqrt2` ....(ii)
For ` z gt gt R`, from (i) and (ii) , we have
`B = mu_(0)/(4 pi) (2 pi IR^(2))/((R^(2) + z^(2))^(3//2)) = mu_(0)/( 4 pi) (2 pi IR^(2))/((z^(2))^(3//2))` ....(iii)
`[R^(2) lt lt z^(2)`, it can be ignored in the denominator]
Also, `B. = mu_(0)/(4 pi) (piIR^(2))/((R^(2) + z^(2))^(3//2)) sqrt2 = mu_(0)/(4 pi) (pi IR^(2) sqrt2)/((z^(2))^(3//2)) ` ...(iv)
From (iii) and (iv) , we have
` B. lt B`
Here, option (a) is correct.