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Q.

Two insulated wires, carrying currents $i_{1} a n d i_{2}$ wrapped over a conical frame form the coils 1 and 2. The distance of the apex from the plane of each ring is much much greater than the radius of the ring. If they produce no net magnetic field at the apex P, the value of $\frac{i_{1}}{i_{2}}$ is

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a

$\sqrt{\frac{r_{2}}{r_{1}}}$

b

$\frac{r_{1}}{r_{2}}$

c

${(\frac{r_{1}}{r_{2}})}^{2}$

d

$\sqrt{\frac{r_{1}}{r_{2}}}$

answer is B.

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Detailed Solution

$\begin{array}{l} B = \frac{μ_{0}}{4 π} \frac{2 M}{x^{3}} = \frac{μ_{0}}{4 π} \frac{{iπr}^{2}}{{(r / t a n θ)}^{3}} {[Each ring behaves like a short magnetic dipole}^{} . \\ \Rightarrow B \propto \frac{i}{r}, ∴ B_{1} = B_{2} \Rightarrow \frac{i_{1}}{r_{1}} = \frac{i_{2}}{r_{2}} \Rightarrow \frac{i_{1}}{i_{2}} = \frac{r_{1}}{{r_{2}}^{}} \end{array}$

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