Question

Give an expression for the magnetic field on the axis of a circular current loop (expression only). What is the value of B at the center of the loop?

Answer 1

Hint: We can derive the expression for magnetic field for variable distance on the axis of a circular current loop by using the Biot-Savart’s law and then to find the magnetic field at the centre, integrate it for zero distance.

Formulae used:
Magnetic field by a circular current carrying conductor at a distance x from the centre on its axis, $B=\dfrac{{\mu}_{0}IR^2}{2(R^2 +x^2)^{3/2}}$, where R is the radius of the loop and x is the distance from the loop

Complete step by step solution:
We have been asked in the question to give an expression of the magnetic field on the axis of a circular current-carrying loop which we can find using the Biot-Savart law.

The expression is, $B=\dfrac{{\mu}_{0}IR^2}{2(R^2 +x^2)^{3/2}}$, where R is the radius of the loop, I is the current flowing through it and x is the distance from the loop.
Now, in order to find the magnetic field at the center, we need to put in the formula the value of x=0
That is, $B=\dfrac{{\mu}_{0}IR^2}{2(R^2 +0^2)^{3/2}}=\dfrac{{\mu}_{0}IR^2}{2(R^2)^{3/2}}=\dfrac{{\mu}_{0}I}{2R}$.
Hence, the magnetic field at the centre of the circular loop will be $B=\dfrac{{\mu}_{0}I}{2R}$.

Additional Information:
Biot-Savart law is the equation which is used to describe the magnetic field generated by a constant electric current relating it to the magnitude, length, direction and the proximity of the electric current. The equation is given by, $B(r)=\dfrac{{{\mu }_{0}}}{4\pi }{{\int }_{C}}\dfrac{I\ dl\times r'}{|r'{{|}^{3}}}$, where dl is the element of wire and a vector along the path C over which the field is needed to be find, l is the point on the path at which the magnetic field is needed to be found, I is the current flowing through the conductor and r is unit vector.

Note: Magnetic field lines around a circular loop are formed as concentric circles around the line of current flow and its radius keeps on increasing as we move from the circumference to the center of the circle where it becomes almost straight. At the center, the magnetic field becomes perpendicular to the plane of the loop.