Name: Conversion of a Galvanometer Into a Voltmeter
Uploaded: 2019-07-19
Duration: 7 min 18 s
Description: This video helps us understand the working of a galvanometer. It also teaches us how to convert it into a voltmeter of the desired range both theoretically a...

Question

Use Biot-Savart law to derive the expression for the magnetic field due to a circular coil of radius $R$ having $N$ turns at a point on the axis at a distance $' x'$ from its centre. How is the direction of the magnetic field determined at this point? Draw the magnetic field lines due to this coil.

Accepted Answer

Let $P$ be a distance $x$ from the center of the loop. From the right-hand rule, the magnetic field $\vec{d B}$ at $P$ , produced by the current element $I \vec{d l}$ , is directed at an angle $ϕ$ above the $y$ -axis as shown. Since $\vec{d l}$ is parallel along the $x$ -axis and $\hat{r}$ is in the $y z$ -plane, the two vectors are perpendicular, so for $N$ such loop we have

$d B = \frac{μ_{0}}{4 π} . \frac{N I d l \sin θ}{s^{2}} = \frac{μ_{0}}{4 π} . \frac{N I d l}{x^{2} + R^{2}}$

where we have used $s^{2} = x^{2} + R^{2}$

The components of $\vec{d B}$ perpendicular to the $y$ -axis cancel each other, and in calculating the net magnetic field, only the components along the $x$ -axis need to be considered. The components perpendicular to the axis of the loop sum to zero in pairs. Hence at point $P$ :

$B = \int_{l o o p} d B \sin ϕ = \int_{l o o p} \frac{μ_{0}}{4 π} . \frac{N I d l \sin ϕ}{x^{2} + R^{2}}$

Here, $\sin ϕ = \frac{R}{s} = \frac{R}{{(x^{2} + R^{2})}^{1 / 2}}$

$∴ B = \frac{μ_{0}}{4 π} \int_{l o o p} \frac{N I R d l}{{(x^{2} + R^{2})}^{3 / 2}}$

The Integral dl over the loop will give us, $2 π R$ .

Hence the magnetic will be equals to

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

Use Biot-Savart law to derive the expression for the magnetic field due to a circular coil of radius $R$ having $N$ turns at a point on the axis at a distance $' x'$ from its centre. How is the direction of the magnetic field determined at this point? Draw the magnetic field lines due to this coil.

Important Questions on Magnetic Effect of Current

Learn and Prepare for any exam you want !

Use Biot-Savart law to derive the expression for the magnetic field due to a circular coil of radius R having N turns at a point on the axis at a distance 'x' from its centre. How is the direction of the magnetic field determined at this point? Draw the magnetic field lines due to this coil.

Important Questions on Magnetic Effect of Current

Learn and Prepare for any exam you want !

Use Biot-Savart law to derive the expression for the magnetic field due to a circular coil of radius $R$ having $N$ turns at a point on the axis at a distance $' x'$ from its centre. How is the direction of the magnetic field determined at this point? Draw the magnetic field lines due to this coil.