Question

If $B_c$ is the magnetic induction at the ceflte ofa circular coit carrying curreflt, then the magnetic induction at a poitrt oo the axis of the coil at a distance equal to the radius of the coil is

• A. $\frac{B_c}{2\sqrt2}$
• B. $\frac{B_c}{2}$
• C. $\frac{B_c}{\sqrt2}$
• D. $\frac{B_c}{4}$
• E. $\frac{B_c}{8}$

Answer 1

The Correct Option is C

Magnetic Induction: Center vs. Axis of a Coil 

We want to find the magnetic induction at a point on the axis of a circular coil, a distance \(r\) (the radius) away from the center, in terms of the magnetic induction at the center of the coil.

Step 1: Magnetic Induction at the Center of a Circular Coil

The magnetic induction (\(B_c\)) at the center of a circular coil of radius \(r\) with \(N\) turns carrying a current \(I\) is given by:

\(B_c = \frac{\mu_0 N I}{2r}\)

Where \(\mu_0\) is the permeability of free space.

Step 2: Magnetic Induction on the Axis of a Circular Coil

The magnetic induction (\(B_{axis}\)) at a point on the axis of the same circular coil, at a distance \(x\) from the center, is given by:

\(B_{axis} = \frac{\mu_0 N I r^2}{2 (r^2 + x^2)^{3/2}}\)

Step 3: Setting \(x = r\)

We are given that the distance from the center is equal to the radius, so we set \(x = r\):

\(B_{axis} = \frac{\mu_0 N I r^2}{2 (r^2 + r^2)^{3/2}}\)

\(B_{axis} = \frac{\mu_0 N I r^2}{2 (2r^2)^{3/2}}\)

\(B_{axis} = \frac{\mu_0 N I r^2}{2 (2^{3/2} r^3)}\)

\(B_{axis} = \frac{\mu_0 N I r^2}{2 (2 \sqrt{2} r^3)}\)

\(B_{axis} = \frac{\mu_0 N I}{4 \sqrt{2} r}\)

Step 4: Relate \(B_{axis}\) to \(B_c\)

We want to express \(B_{axis}\) in terms of \(B_c = \frac{\mu_0 N I}{2r}\). Notice that \(\frac{\mu_0 N I}{r} = 2B_c\):

\(B_{axis} = \frac{1}{2\sqrt{2}} \cdot \frac{\mu_0 N I }{2r} \frac{2}{1}\)

\(B_{axis} = \frac{1}{2\sqrt{2}} \cdot \frac{\mu_0 N I }{r}\)

\(B_{axis} = \frac{1}{2\sqrt{2}} \cdot {2 B_c}\)

\(B_{axis} = \frac{B_c}{\sqrt{2}}\)

Conclusion

The magnetic induction at a point on the axis of the coil at a distance equal to the radius of the coil is not any of the solution set but follows to be \(\frac{B_c}{\sqrt{2}}\)