Biot-Savart law $XY$ is a current carrying wire.$\overset{\rightarrow}{d l}$ is a small element on it.At point $P$, whose position vector is $\overset{\rightarrow}{r}$, the magnetic field is to be determined.According to Biot Savart law, the magnitude of magnetic field $\overset{\rightarrow}{d B}$ at $P$ is(i) Proportional to current I(ii) Proportional to length dl(iii) Inversely proportional to the square of the distance of the pointThe direction of magnetic field is perpendicular to the plane containing $\overset{\rightarrow}{d l}$ and $\overset{\rightarrow}{r}$.In vector form, $\;\overset{\rightarrow}{d B} \propto \;\frac{\overset{\rightarrow}{I d l \overset{\rightarrow}{r}}}{r^{3}}$ Or, $\;\overset{\rightarrow}{d B} = \frac{\mu_0}{4\pi} \;\frac{\overset{\rightarrow}{I d l \overset{\rightarrow}{r}}}{r^{3}}$ Magnetic field due to a current carrying circular coil: A single turn circular coil of radius a carrying current $I$ is considered. $P$ is a point on the axis at a distance $d$ where the magnetic field is to be determined. Two small lengths $dl$ are considered at two diametrical opposite ends on the coil. Distance of point $P$ from $dl$ is $r$. If $dB$ is the magnetic field, then $d B = \frac{\mu_0}{4\pi} \frac{I d l \sin 90^{\circ}}{r^{2}}$ $= \frac{\mu_0}{4\pi} \frac{I d l}{r^{2}}$ The 2 components of $d B$ are $d B \cos \phi$ and $d B \sin \phi$. The two $d B \cos \phi$ components corresponding to two dl elements (at the upper and the lower end) cancel each other. The two $d B \sin \phi$ components are in same direction and hence resultant magnetic field at $P$ becomes $2 d B \sin \phi$. So, the resultant magnetic field at point $P$ due to the entire coil is $B = \frac{\mu_0}{4\pi} \Sigma \frac{2 I d l \sin \phi}{r^{2}}$ Or, $B = \frac{\mu_0}{4\pi} \frac{2 I \sin \phi}{r^{2}} \Sigma d l$ Or, $B = \frac{\mu_0}{4\pi} \frac{2 I \sin \phi}{r^{2}} \times \pi a$ [since at a time two dl portions have been considered at two diametrical opposite ends.] Or, $B = \frac{\mu_0}{4\pi} \frac{2 I \times \frac{a}{r}}{r^{2}} \times \pi a$[$\text{ since, } \sin \phi = \frac{a}{r} \text{ ] }.$ Or, $B = \frac{\mu_0}{4\pi} \frac{2 \pi a^{2} I}{r^{3}}$ Or, $B = \frac{\mu_0}{2} \frac{a^{2} I}{(a^{2}+d^{2})^{3/2}}$Ratio of magnetic fields: When $d = a\sqrt{3}$ $B_P = \frac{\mu_0}{2} \frac{a^{2} I}{[a^{2}+(a\sqrt{3})^{2}]^{3/2}}$ $= \frac{\mu_0}{2} \times \frac{I}{8a}$At centre $(d=0)$ $B_{\text{centre}} = \frac{\mu_0}{2} \times \frac{a^{2} I}{(a^{2})^{3/2}}$ $= \frac{\mu_0}{2} \times \frac{I}{a}$So, the ratio $= \frac{B_{\text{centre}}}{B_p} = 8:1$ 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 $L$ and breadth $b$, having $n$ number of turns, current $i$ 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 $F = n B i L$. These two forces are oppositely directed. So, the moment of deflecting couple is $\tau = f \times b = n \text{ Bilb }$ As the coil rotates, a restoring torque $c\theta$ is produced in the phosphor bronze strip, where c is the torsional constant and $\theta$ is the angle of twist. At equilibrium, $\text{ Or, } c\theta = n B i A \quad (A = \text{ area of the coil })$ $\text{ Or, } i = \frac{c\theta}{n B A}$ $\text{ Or, } i = k \theta$ $\left(k = \frac{c}{n B A} = \text{ Galvanometer constant }\right)$ $\therefore i \propto \theta$ 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.