Biot-Savart law
$\mathrm{XY}$ is a current carrying wire.
$\overrightarrow{\mathit{d}\mathit{l}}$ is a small element on it.
At point $\mathrm{P}$, whose position vector is $\overrightarrow{\mathit{r}}$, the magnetic field is to be determined.
According to Biot Savart law, the magnitude of magnetic field $\overrightarrow{\mathit{d}\mathit{B}}$ at $\mathrm{P}$ is
(i) Proportional to current I
(ii) Proportional to length dl
(iii) Inversely proportional to the square of the distance of the point
The direction of magnetic field is perpendicular to the plane containing $\overrightarrow{\mathit{d}\mathit{l}}$ and $\overrightarrow{\mathit{r}}$.
In vector form,
$\overrightarrow{\mathit{d}\mathit{B}}\propto \frac{\mathit{I}\overrightarrow{\mathit{d}\mathit{l}}\mathit{X}\overrightarrow{\mathit{r}}}{{\mathit{r}}^3}$
Or, $\overrightarrow{\mathit{d}\mathit{B}}=\frac{{\mathit{\mu }}_0}{4\mathit{\pi }}\frac{\mathit{I}\overrightarrow{\mathit{d}\mathit{l}}\mathit{X}\overrightarrow{\mathit{r}}}{{\mathit{r}}^3}$
Magnetic field due to a current carrying circular coil:
A single turn circular coil of radius a carrying current $\mathrm{I}$ is considered. $\mathrm{P}$ is a point on the axis at a distance $\mathrm{d}$ where the magnetic field is to be determined.
Two small lengths $\mathrm{dl}$ are considered at two diametrical opposite ends on the coil.
Distance of point $\mathrm{P}$ from $\mathrm{dl}$ is $\mathit{r}$.
If $\mathrm{dB}$ is the magnetic field, then
$\mathit{d}\mathit{B}=\frac{{\mathit{\mu }}_0}{4\mathit{\pi }}\frac{\mathit{I}\mathit{d}\mathit{l}\mathit{s}\mathit{i}\mathit{n}{90}^{\circ }}{{\mathit{r}}^2}$
$=\frac{{\mathit{\mu }}_0}{4\mathit{\pi }}\frac{\mathit{I}\mathit{d}\mathit{l}}{{\mathit{r}}^2}$
The 2 components of $\mathit{d}\mathrm{B}$ are $\mathit{d}\mathrm{B}\cos \mathit{\varphi }$ and $\mathit{d}\mathrm{B}\mathit{s}\mathit{i}\mathit{n}\mathit{\varphi }$.
The two $\mathit{d}\mathrm{B}\cos \mathit{\varphi }$ components corresponding to two dl elements (at the upper and the lower end) cancel each other.
The two $\mathit{d}\mathrm{B}\mathit{s}\mathit{i}\mathit{n}\mathit{\varphi }$ components are in same direction and hence resultant magnetic field at $\mathrm{P}$ becomes $2\mathit{d}\mathrm{B}\mathit{s}\mathit{i}\mathit{n}\mathit{\varphi }$.
So, the resultant magnetic field at point $\mathrm{P}$ due to the entire coil is
$\mathit{B}=\frac{{\mathit{\mu }}_0}{4\mathit{\pi }}\mathit{\Sigma }\frac{2\mathit{I}\mathit{d}\mathit{l}\mathit{s}\mathit{i}\mathit{n}\mathit{\varphi }}{{\mathit{r}}^2}$
Or, $\mathit{B}=\frac{{\mathit{\mu }}_0}{4\mathit{\pi }}\frac{2\mathit{I}\mathit{s}\mathit{i}\mathit{n}\mathit{\varphi }}{{\mathit{r}}^2}\mathit{\Sigma }\mathit{d}\mathit{l}$
Or, $\mathit{B}=\frac{{\mathit{\mu }}_0}{4\mathit{\pi }}\frac{2\mathit{I}\mathit{s}\mathit{i}\mathit{n}\mathit{\varphi }}{{\mathit{r}}^2}\times \mathit{\pi }\mathit{a}$
[since at a time two dl portions have been considered at two diametrical opposite ends.]
Or, $\mathit{B}=\frac{{\mathit{\mu }}_0}{4\mathit{\pi }}\frac{2\mathit{I}\times \frac{\mathit{a}}{\mathit{r}}}{{\mathit{r}}^2}\times \mathit{\pi }\mathit{a}$
[
Or, $\mathit{B}=\frac{{\mathit{\mu }}_0}{4\mathit{\pi }}\frac{2\mathit{\pi }{\mathit{a}}^2\mathit{I}}{{\mathit{r}}^3}$
Or, $\mathit{B}=\frac{{\mathit{\mu }}_0}{2}\frac{{\mathit{a}}^2\mathit{I}}{{\left({\mathit{a}}^2+{\mathit{d}}^2\right)}^{3∕2}}$

Ratio of magnetic fields:


When $\mathit{d}=\mathit{a}\sqrt{3}$
${\mathit{B}}_{\mathit{P}}=\frac{{\mathit{\mu }}_0}{2}\frac{{\mathit{a}}^2\mathit{I}}{{\left[{\mathit{a}}^2+{\left(\mathit{a}\sqrt{3}\right)}^2\right]}^{3∕2}}$
$=\frac{{\mathit{\mu }}_0}{2}\times \frac{\mathit{I}}{8\mathit{a}}$

At centre $\left(\mathit{d}=0\right)$

$=\frac{{\mathit{\mu }}_0}{2}\times \frac{\mathit{I}}{\mathit{a}}$
So, the ratio
OR
(a) Magnetic field; lines due to current carrying loop:(b) Working principle of Moving coil galvanometer:

PQRS is a rectangular coil, of copper wire of length $\mathrm{L}$ and breadth $\mathit{b}$, having $\mathit{n}$ number of turns, current $\mathit{i}$ flowing through it, is hung in a permanent magnetic field B with the help of a phosphor bronze strip. Force acting on PQ and SR is $\mathit{F}=\mathit{n}\mathit{B}\mathit{i}\mathit{L}$. These two forces are oppositely directed.
So, the moment of deflecting couple is

As the coil rotates, a restoring torque $\mathit{c}\mathit{\theta }$ is produced in the phosphor bronze strip, where c is the torsional constant and $\mathit{\theta }$ is the angle of twist.
At equilibrium,




$\therefore \mathit{i}\propto \mathit{\theta }$

Function of radial magnetic field: Due to radial magnetic field, magnetic field lines become perpendicular to magnetic moment and hence the torque becomes maximum.
Function of soft iron core: Using soft iron core sensitivity increases since the magnetic field lines prefer to pass through soft iron.
(c) A shunt resistance of small value is connected in parallel with a galvanometer to convert it to an ammeter. Ammeter is used in series in a circuit. Its resistance should be as low as possible so that it does not make any change in the circuit current. So, a low resistance is connected in parallel with the galvanometer to achieve this.
A high value resistance is connected in series with a galvanometer to convert it to an voltmeter. Voltmeter is used in parallel to a component in a circuit. Its resistance should be as high as possible so that it does not make any change in the circuit current. So, a high value resistance is connected in series with the galvanometer to achieve this.