MHT CET · Physics · Magnetic Effects of Current
A current carrying circular coil of radius \(R\) has a point \(P\) situated on its axis at a distance \(x\) from its centre \(\mathrm{O}\) of the coil. The magnetic induction at point \(P\) is \(\left(\frac{1}{8}\right)^{\text {th }}\) of magnetic field at its centre O. The value of \(x\) is
- A \(\frac{R}{2 \sqrt{3}}\)
- B \(\sqrt{3} R\)
- C \(\frac{R}{\sqrt{3}}\)
- D \(\frac{2}{\sqrt{3}} R\)
Answer & Solution
Correct Answer
(B) \(\sqrt{3} R\)
Step-by-step Solution
Detailed explanation
Magnetic field on the axis of a circular coil, at a location \(x\) away from the center of the coil is given by,
\(B=\frac{\mu_0 I R^2}{2\left(R^2+x^2\right)^{\frac{3}{2}}}\)
For, \(x=0\) the induction at the centre of the coil is \(B_0=\left(\frac{\mu_0 I}{2 R}\right)\)
\(\begin{aligned}
& \therefore \frac{1}{8} \times\left(\frac{\mu_0 I}{2 R}\right)=\frac{\mu_0 I R^2}{2\left(R^2+x^2\right)^{\frac{3}{2}}} \\
& \Rightarrow(2 R)^3=\left(R^2+x^2\right)^{\frac{3}{2}} \\
& \Rightarrow(2 R)=\left(R^2+x^2\right)^{\frac{1}{2}} \\
& \Rightarrow(2 R)^2=\left(R^2+x^2\right) \\
& \Rightarrow x=\sqrt{3} R
\end{aligned}\)
\(B=\frac{\mu_0 I R^2}{2\left(R^2+x^2\right)^{\frac{3}{2}}}\)
For, \(x=0\) the induction at the centre of the coil is \(B_0=\left(\frac{\mu_0 I}{2 R}\right)\)
\(\begin{aligned}
& \therefore \frac{1}{8} \times\left(\frac{\mu_0 I}{2 R}\right)=\frac{\mu_0 I R^2}{2\left(R^2+x^2\right)^{\frac{3}{2}}} \\
& \Rightarrow(2 R)^3=\left(R^2+x^2\right)^{\frac{3}{2}} \\
& \Rightarrow(2 R)=\left(R^2+x^2\right)^{\frac{1}{2}} \\
& \Rightarrow(2 R)^2=\left(R^2+x^2\right) \\
& \Rightarrow x=\sqrt{3} R
\end{aligned}\)
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