First slide

Magnetic Dipole moment

Question

A small ring of radius ‘a’ is carrying i, P is a point on its axis at a large distance x from its center. Mathematic induction at P is found to be B₁. If the radius is doubled and current is also doubled. Also the distance of P from the center of the ring is doubled. Then find new magnetic induction at P.

Moderate

A

2B
B

4B
C

B
D

B/2

Solution

The current carrying ring behaves like a short magnetic dipole of dipole moment M = πa² i.
$∴ B_{P} = \frac{μ_{o}}{4 π} \frac{2 M}{x^{3}}$
Initially B = $\frac{μ_{0}}{4 π} \frac{2 i π a^{2}}{x^{3}}$
In the second condition $B_{2} = \frac{μ_{0}}{4 π} \frac{2 π (4 a^{2}) (2 i)}{{(2 x)}^{3}} = \frac{μ_{o}}{4 π} \frac{2 i a^{2} π}{x^{3}}$
Thus new magnetic induction = B

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