Biot-Savart law
XY is a current carrying wire.

→

dl

is a small element on it.
At point P, whose position vector is

→

r

, the magnetic field is to be determined.
According to Biot Savart law, the magnitude of magnetic field

→

dB

at 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

→

dl

and

→

r

.
In vector form,
‌

→

dB

∝‌

I → dl X → r

r3

Or, ‌

→

dB

=‌

µ0

4π

‌

I → dl X → r

r3

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
dB‌=‌

µ0

4π

‌

Idlsin‌90∘

r2

‌=‌

µ0

4π

‌

Idl

r2

The 2 components of dB are dB‌cos‌ϕ and dBsin‌ϕ.
The two dB‌cos‌ϕ components corresponding to two dl elements (at the upper and the lower end) cancel each other.
The two dBsin‌ϕ components are in same direction and hence resultant magnetic field at P becomes 2dBsin‌ϕ.
So, the resultant magnetic field at point P due to the entire coil is
‌B=‌

µ0

4π

Σ‌

2Idlsin‌ϕ

r2

Or, ‌B=‌

µ0

4π

‌

2Isin‌ϕ

r2

Σdl
Or, ‌B=‌

µ0

4π

‌

2Isin‌ϕ

r2

×πa
[since at a time two dl portions have been considered at two diametrical opposite ends.]
Or, ‌B=‌

µ0

4π

‌

2I× a r

r2

×πa
[‌ since, ‌sin‌ϕ=‌

a

r

‌ ] ‌.
Or, ‌B=‌

µ0

4π

‌

2πa2I

r3

Or, ‌B=‌

µ0

2

‌

a2I

(a2+d2)3∕2

Ratio of magnetic fields:


When ‌d=a√3
‌BP=‌

µ0

2

‌

a2I

[a2+(a√3)2]3∕2

‌=‌

µ0

2

×‌

I

8a

At centre (d=0)
‌B‌centre ‌=‌

µ0

2

×‌

a2I

(a2)3∕2

‌=‌

µ0

2

×‌

I

a

So, the ratio =‌

B‌centre ‌

Bp

=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=nBiL. These two forces are oppositely directed.
So, the moment of deflecting couple is
τ=f×b=n‌ Bilb ‌
As the coil rotates, a restoring torque cθ is produced in the phosphor bronze strip, where c is the torsional constant and θ is the angle of twist.
At equilibrium,
‌ Or, ‌‌cθ‌=nBiA‌‌(A=‌ area of the coil ‌)
‌ Or, ‌‌i=‌

cθ

nBA

‌ Or, i=kθ
(k=‌

c

nBA

=‌ Galvanometer constant ‌)
∴‌i∝θ

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.