First slide

Magnetic field due to a current carrying circular ring

Question

Two concentric coplanar circular loops of radii r₁ and r₂ cany currents i₁ and i₂, respectively, in opposite directions (one clockwise and the other anticlockwise.) The magnetic induction at the centre of the loops is half that due to i₁ alone at the centre. If r₂ = 2r₁ , the value of i₂/i₁ is

Moderate

A

2
B

1/2
C

1/4
D

1

Solution

Magnetic field at the centre due to smaller loop
$B_{1} = \frac{μ_{0}}{4 π} . \frac{2 {πi}_{1}}{r_{1}}$ ….(i)
Due to bigger loop, $B_{2} = \frac{μ_{0}}{4 π} . \frac{2 {πi}_{2}}{r_{2}}$
So, net magnetic field at centre
$B = B_{1} - B_{2} = \frac{μ_{0}}{4 π} \times 2 π (\frac{i_{1}}{r_{1}} - \frac{i_{2}}{r_{2}})$
According to question, $B = \frac{1}{2} \times B_{1}$
$\begin{array}{l} \Rightarrow \frac{μ_{0}}{4 π} . 2 π (\frac{i_{1}}{r_{1}} - \frac{i_{2}}{r_{2}}) = \frac{1}{2} \times \frac{μ_{0}}{4 π} . \frac{2 {πi}_{1}}{r_{1}} \\ \frac{i_{1}}{r_{1}} - \frac{i_{2}}{2 r_{1}} = \frac{i_{1}}{2 r_{1}} \Rightarrow \frac{i_{1}}{i_{2}} = 1 \{∵ r_{2} = 2 r_{1}\} \end{array}$

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