Name: Magnetic Flux
Uploaded: 2019-07-19
Duration: 6 min 44 s
Description: Magnetic flux is the total number of magnetic lines of force that are passing through a given surface. To learn more about this intriguing concept, watch the...

Question

State Biot and Savart law. Derive the expression for the magnetic field at any point on the axis of a circular current-carrying loop. Find its value at the centre of this loop.

Accepted Answer

The Biot Savart Law is used to determine the magnetic field intensity $B$ near a current-carrying conductor in terms of magnetic field intensity generated by its source current element.
Consider a wire carrying an electric current $I$ and also consider an infinitely small length of a wire $d l$ at a distance $r$ from point $A$ .

Biot Savart Law states that magnetic intensity $d B$ at a point $A$ due to current I flowing through a small element $d l$ is
Directly proportional to current
Directly proportional to the length of the element $(d l)$
Directly proportional to the sine of angle $θ$ between the direction of current and the line joining the element $d l$ from point $A$ .
Inversely proportional to the square of the distance of point $A$ from the element $d l$ .

$d B \propto \frac{I d l \sin θ}{r^{2}}$

$d B = \frac{μ_{0}}{4 π} \frac{I d l \sin θ}{r^{2}}$

$μ_{0}$ =Absolute permittivity of air in a vacuum.

Let $O$ be the centre of the ring, $R$ be the radius of the ring. The point $P$ lie outside the ring on the axis, let the distance between the $P$ and $O$ be $x$ , where we have to calculate the magnetic field.

Let $d B$ be the magnetic field due to a small length dl of the ring,

As per biot-savart rule,

$d B = \frac{μ_{0} I d l}{4 π r^{2}}$

We know that $r = \sqrt{x^{2} + R^{2}}$

This field has vertical component $d B_{y} = d B \sin ϕ$ and horizontal component $d B_{x} = d B \cos ϕ$

The vertical component will cancel out each other, only horizontal components are responsible.

So, $B = d B_{1} + d B_{2} + . . . . .$

$d B = \frac{μ_{0} I d l}{4 π r^{2}} \sin ϕ$

$d B = \frac{μ_{0} I d l}{4 π r^{2}} \times \frac{R}{r}$

$d B = \frac{μ_{0} I R}{4 π (x^{2} + R^{2})^{\frac{3}{2}}} d l$

Total field $B = \int d B = \int \frac{μ_{0} I R}{4 π (x^{2} + R^{2})^{\frac{3}{2}}} d l$

$B = \frac{μ_{0} I R}{4 π (x^{2} + R^{2})^{\frac{3}{2}}} \int d l$

But $\int d l =$ Length of the loop $= 2 π R$

$B = \frac{μ_{0} I R}{4 π (x^{2} + R^{2})^{\frac{3}{2}}} \times 2 π R = \frac{μ_{0} I R^{2}}{2 (x^{2} + R^{2})^{\frac{3}{2}}}$

At the centre of the loop $x = 0$

By putting this value in the above equation

$B = \frac{μ_{0} I R^{2}}{2 (R^{2})^{\frac{3}{2}}} = \frac{μ_{0} I}{2 R}$

State Biot and Savart law. Derive the expression for the magnetic field at any point on the axis of a circular current-carrying loop. Find its value at the centre of this loop.

Important Questions on Magnetic Effect of Electric Current

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