Question

A circular coil of 200 turns has a radius of 10 cm and carries a current of 2.0 A. (a) Find the magnitude of the magnetic field →B at the centre of the coil. (b) At what distance from the centre along the axis of the coil will the field B drop to half its value at the centre?
(3√4=1·5874...)

Answer 1

magnitude of magnetic field at the center of the coil is 2.5*10^-3 T

distance on the axis at which magnetic field becomes half of is value at center is 7.7 cm

The problem is solve as in the picture.

where,

μo = permeability of free space,

N = no of turns in the coil

i = current in the coil

R = radius of the coil

r = distance from the center of the coil in the axial direction.

we solve for magnetic field at the center first. using this value, we find the distance r by the general formula      

Answer 2

(a) The magnitude of the magnetic field →B at the centre of the coil is $12.56 \mathrm{mT}$

(b) At  7.66cm  distance from the centre along the axis of the coil, the field B drop to half its value at the centre.

Explanation:

Given:

Coil turns in loop, n = 200 turns

coil Radius, r = 10 cm

coil Current, i = 2 A

(a) Let the coil's magnetic field at its core be B.

As the magnetic field relationship at the center of a circular coil is given

$B=\frac{n \mu_{0} i}{2 r}$

  $=\frac{200 \times 4 \pi \times 10^{-7} \times 2}{2 \times 10 \times 10^{-2}}$

  $=2.51 \times 10^{-3}$

  $=12.56 \mathrm{mT}$

(b) As magnetic field on the axis of the rotating coil at any point P (say) is given by

$B_{p}=\frac{n \mu_{0} i r^{2}}{2\left(x^{2}+r^{2}\right)^{\frac{3}{2}}}$

Where x is the point distance from coil core.  

As the question relates

$\frac{1}{2} B_{\text {centre }}=B_{p}$

$\frac{1}{2} \frac{n \mu_{0} i}{2 r}=\frac{n \mu_{0} i r^{2}}{2\left(x^{2}+r^{2}\right)^{\frac{3}{2}}}$

$\left(x^{2}+r^{2}\right)^{\frac{3}{2}}=2 r^{3}$

$x^{2}+r^{2}=4^{\frac{1}{3}} r^{2}$

$x^{2}+r^{2}=1.58 r 2$

x = 0.766

$x=\pm 7.66 \mathrm{cm}$

The magnetic field will decrease to half its value in the middle if the distance from the middle of the spiral along the spiral axis is equal to 7.66 cm