$\frac{B_c}{\sqrt2}$
$\frac{B_c}{4}$
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.
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.
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:
\(B_{axis} = \frac{\mu_0 N I r^2}{2 (r^2 + x^2)^{3/2}}\)
We are given that the distance from the center is equal to the radius, so we set \(x = r\):
\(B_{axis} = \frac{\mu_0 N I r^2}{2 (r^2 + r^2)^{3/2}}\)
\(B_{axis} = \frac{\mu_0 N I r^2}{2 (2r^2)^{3/2}}\)
\(B_{axis} = \frac{\mu_0 N I r^2}{2 (2^{3/2} r^3)}\)
\(B_{axis} = \frac{\mu_0 N I r^2}{2 (2 \sqrt{2} r^3)}\)
\(B_{axis} = \frac{\mu_0 N I}{4 \sqrt{2} r}\)
We want to express \(B_{axis}\) in terms of \(B_c = \frac{\mu_0 N I}{2r}\). Notice that \(\frac{\mu_0 N I}{r} = 2B_c\):
\(B_{axis} = \frac{1}{2\sqrt{2}} \cdot \frac{\mu_0 N I }{2r} \frac{2}{1}\)
\(B_{axis} = \frac{1}{2\sqrt{2}} \cdot \frac{\mu_0 N I }{r}\)
\(B_{axis} = \frac{1}{2\sqrt{2}} \cdot {2 B_c}\)
\(B_{axis} = \frac{B_c}{\sqrt{2}}\)
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 \(\frac{B_c}{\sqrt{2}}\)
The magnetic induction at a point on the axis of a circular coil at a distance \( r \) from the centre (where \( r \) is also the radius of the coil) is given by: \[ B = \frac{B_c}{\sqrt{2}} \] Here, \( B_c \) is the magnetic induction at the centre of the coil. The magnetic induction decreases with distance from the centre, and at a point on the axis at a distance equal to the radius of the coil, it is \( \frac{B_c}{\sqrt{2}} \).