Name: Motion in Combined Electric and Magnetic Fields
Uploaded: 2019-07-19
Duration: 6 min 38 s
Description: Have you ever thought about the motion of charged particle when it passes through the combination of the electric and magnetic field? Let's watch this video...

Question

Derive an expression for the magnetic induction at a point on the axis of a current carrying circular coil using Biot-Savart law.

Accepted Answer

Let a point P on the axis of the circular coil carrying current $I$ , where we have to find the magnetic field. The loop shown above is in the y-z plane with its centre at the origin and has a radius $R$ . The $x -$ axis is the axis of the loop. Let $x$ be the distance of point P from the centre. Consider a conducting element $d l$ of the loop. The magnitude $d B$ of the magnetic field due to $d l$ is given by the Biot-Savart law:

$d B = \frac{μ_{0}}{4 π} \frac{I |\vec{d l} \times \vec{r}|}{r^{3}}$ ......(i)

Now using Pythagoras theorem:

$r^{2} = x^{2} + R^{2}$ .....(ii)

Any element of the conductor is perpendicular to the displacement vector. Hence:

$|\vec{d l} \times \vec{r}|$ $= r d l$ .....(iii)

Using (i), (ii) and (iii), the magnetic field due to current element:

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

The direction of $d B$ is perpendicular to the plane formed by $d l$ and $r$ . Its components perpendicular to x-axis cancel out. Only the $x -$ components survive. Hence, the net contribution along $x -$ axis:

$d B_{x} = d B \cos θ$ ......(v)

$cosθ = \frac{R}{{(x^{2} + R^{2})}^{\frac{1}{2}}}$ .....(vi)

Using (iv), (v) and (vi) we get:

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

The summation of elements $d l$ along the circumference of the loop gives $2 π R$ . Thus, the field at P due to entire loop is:

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

Derive an expression for the magnetic induction at a point on the axis of a current carrying circular coil using Biot-Savart law.

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