For a circular coil of radius R and N turns carrying current I, the magnitude of the magnetic field at a point on its axis at a distance x from its centre is given by,
$B = \frac{μ_{0} {IR}^{2} N}{2 {(x^{2} + R^{2})}^{3 / 2}}$
(a) Show that this reduces to the familiar result for field at the centre of the coil.

(b) Consider two parallel co-axial circular coils of equal radius R, and number of turns N, carrying equal currents in the same direction, and separated by a distance R. Show that the field on the axis around the mid-point between the coils is uniform over a distance that is small as compared to R, and is given by,
$B = 0.72 \frac{μ_{0} NI}{R}, approximately .$
[Such an arrangement to produce a nearly uniform magnetic field over a small region is known as Helmholtz coils.]

Solution

(a) Given, a circular coil of radius r and N turns carrying current I.

Then,
Magnitude of magnetic field at a point on axis at a distance x from centre is given by,
   $B = \frac{μ_{0} {IR}^{2} N}{2 {(x^{2} + R^{2})}^{3 / 2}} (axial line)$

At the centre of the coil x =0

Therefore,

   $B = \frac{μ_{0} {IR}^{2} N}{2 R^{3}}$

i.e.,    $B = \frac{μ_{0} IN}{2 R}$

which is same as the standard result.

(b) In figure, O is a point which is mid-way between the two coils X and Y.

Let, B_x be the magnetic field at Q due to coil X.
Then,
$B_{x} = \frac{μ_{0} {NIR}^{2}}{2 {[{(\frac{R}{2} + d)}^{2} + R^{2}]}^{3 / 2}}$

If, $B_{y}$ is the magnetic field at Q due to coil Y, then
   $B_{y} = \frac{μ_{0} {NIR}^{2}}{2 {[{(\frac{R}{2} - d)}^{2} + R^{2}]}^{3 / 2}}$

The currents in both the coils X and Y are flowing in the same direction.

So, the resultant field is given by

Some More Questions From Moving Charges And Magnetism Chapter

Two moving coil meters, M₁ and M₂ have the following particulars:
R₁ = 10 Ω N₁ = 30,
A₁ = 3.6 x 10^–3 m², B₁ = 0.25 T
R₂ = 14 Ω; N₂ = 42,
A₂ = 1.8 x 10^–3 m², B₂ = 0.50 T
(The spring constants k are identical for the two meters).
Determine the ratio of (a) current sensitivity and (b) voltage sensitivity of M₂ and M_1.

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Some More Questions From Moving Charges And Magnetism Chapter

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