Question

Obtain an expression for magnetic field on the axis of a current carrying circular loop (coil) with the help of Biot-savart law. Draw necessary diagram.

Accepted Answer

Let current carrying circular loop of radius $R$ , carrying current $I$ is placed in $y z$ plane and point $P$ is place on axis of the loop (i.e. $x$ -axis) at a distance $x$ from centre of loop $O$ as shown in figure.

Consider element $d l$ of loop as shown in the figure. Let magnetic field due to this element is $d B$ .

By applying the Biot-Savart rule,

$d B = \frac{μ_{\circ}}{4 π} \frac{I | d l \times r |}{r^{3}} = \frac{μ_{\circ}}{4 π} \frac{I d l s i n ϕ}{r^{2}}$

where $ϕ$ is the angle between $d l$ and the displacement vector $r$

Since the point is on the axis of the loop, therefore, displacement vector $r$ is always perpendicular to the element on the loop $d l$

$∴ ϕ = 90 ° \Rightarrow \sin ϕ = 1$

By Pythagoras theorem, $r^{2} = x^{2} + R^{2}$

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

As we see in the figure magnetic field $d B$ has component $d B_{x}$ along $x -$ axis and component $d B_{y}$ along $y -$ axis

$d B_{y} = d B \sin θ$ and $d B_{x} = d B \cos θ$ ,

where $θ$ is angle made by $d B$ vector and with axis of loop ( $x -$ axis)

The net sum of component of the magnetic field along $y -$ axis will be zero. Hence, net electric field due to all elements of the loop at point $P$ will be equal to the sum of component of the magnetic field along $x -$ axis

$∴ B = \int_{0}^{2 π R} d B \cos θ = \int_{0}^{2 π R} \frac{μ_{\circ}}{4 π} \frac{I \cos θ}{(x^{2} + R^{2})} d l = \frac{μ_{\circ}}{4 π} \frac{I co s θ}{(x^{2} + R^{2})} \int_{0}^{2 π R} d l$

$\Rightarrow B = \frac{μ_{\circ} I R \cos θ}{2 (x^{2} + R^{2})} \hat{i}$

Now, $\cos θ = \frac{R}{{(x^{2} + R^{2})}^{1 / 2}}$

$∴ B = \frac{μ_{\circ} I R^{2}}{2 {(x^{2} + R^{2})}^{3 / 2}} \hat{i}$

Hence, the expression for the magnetic field due to the current carrying coil is $\frac{μ_{o} I R^{2}}{2 {(x^{2} + R^{2})}^{3 / 2}}$ .

Obtain an expression for magnetic field on the axis of a current carrying circular loop (coil) with the help of Biot-savart law. Draw necessary diagram.

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