## Magnetic Effects of Current and Magnetism

Force on a conductor carrying a current placed in a magnetic filed ( F ) –

F = i (I x B )

Or F = i / B sin θ

Where F = magnitude of the force on the conductor (in Newton)

i = current in ampere flowing through conductor

*I* = length of the conductor (in metre)

B = magnetic field at the place of conductor (in Wb/m^{2} or Tesla)

and θ = angle between the magnetic field and the length of the conductor

**Force on a moving charge in a magnetic field –**

F = Q (v x B)

Or F = QvB sin θ

Where Q = charge (in coulomb)

V = velocity of moving charge (in metre/second)

**Lorentz equation for force (F) between moving charges:**

F = q E + q ( v x B)

Where E and B are the electric and magnetic field strengths respectively due to one charge and V is the velocity of the moving charge q.

**Ampere’s swimming rule** – It state that if a person is imagined to swim above the conductor with his face downwards and in the direction of the current, hen the north pole of the needle placed below the conductor gets deflected towards his left hand.

**Maxwells’ cork screw rule** – If a right handed cork screw is imagined to be rotated in such a direction that the tip of the screw advances in the direction of the current, then the direction of rotation of thumb gives the direction of magnetic field.

**Right hand rule** – If a conductor carrying current is grasped in the right hand such that the thumb points in the direction of the current, then the direction of the curl of the rest of fingers gives the directions of the magnetic field.

**Biot-Savart Law (or Laplace’s Law)** – I states that, the magnetic field ‘dB’ at a point due to the current element of length dt is given by –

where I = current flowing in the conductor

r = distance of the point from the

conductor

μ = absolute permeability of the medium

**Magnetic field due to a straight infinite conductor –**

Where i = current in ampere

Force between two parallel straight wires carrying currents –

Where i_{1} & i_{2}= current flowing in two conductors respectively

r = distance between two conductors

and l = length of the conductor under consideration

**Unite of current (ampere)** – The expression for force helps us to define an ampere as follows “The electric current which when flowing in each of the two infinitely long parallel straight conductors of negligible cross-section placed I m apart in empty space makes them exert a force of 2 x 10^{–7} newtons per metre of length”.

**Magnetic filed due to a circular coil –**

(i) At centre of the coil –

(ii) At the axis of the coil –

Where n = number of turn in coil

i = current flowing in the coil

a = radius of the circular coil

And x = distance of the point on the axis, from the centre of the coil.

**Magnetic field due to a long solenoid –**

(i) At any point on the axis–

(ii) At any point well inside a long solenoid –

B = μ_{o} n i

(iii) At ends of the solenoid –

Where n = number o turns per unit length

i = current in ampere

θ_{1} & θ_{2} = angles subtended at the point of consideration by the first and last turn of th solenoid.

**Moving coil galvanometer** – Current (i) in moving coil galvanometer is –

where K = = Instrument constant

C = Torsional rigidity of the suspension wire

N = Number of turns in the coil

B = Magnetic filed induction due to permanent magnet

A = Area of the coil

Φ = Deflection of the coil in radians

**Cyclotron** – is a device used to accelerate charged particles to very high energies.

**Cyclotron radius or gyro-radius** (r)–

where m = mass of the charged particle

v = component of velocity of the charged particle perpendicular to the magnetic field.

B = magnetic field

q = charge on the particle

**Cyclotron Frequency or gyro frequency (f) –**

**Magnetic Poles** – are the parts of the magnet towards which the external magnetizing force tends to converge (south pole) or from which it tends to diverge (north pole).

**Coulomb’s law of force** – States that the force exerted by an isolated magnetic pole of strength m_{1} on another pole of strength m_{2} in free space is directly proportional to the product of their strengths, and inversely proportional to the square of the distance r between them.

Mathematically

Where K = = 10^{–7} weber per ampere – metre

and μ_{0} = permeability of the free space.

**Unit magnetic pole** – is that imaginary pole which when placed in air at a distance of one unit length from an equal and a similar pole repels it with a force of one unit.

**Pole strength – (m)** – The force on the poles per unit filed of induction is called the pole strength.

**Magnetic field** – The space around a magnet (or a current carrying conductor) in which the magnetic effect can be felt is called the magnetic field.

**Intensity of Magnetic field (H)** – It is numerically equal to the force which a unit north pole would experience if placed at that point, it being assumed that unit pole itself does not affect the magnetic field. The S. I. unit of H is ampere per metre (A/m).

**Magnetic field vector (B)**– The magnetic filed vector or the magnetic flux density of the magnetic induction B is related to H as B = μ_{0}. H. Its S. I. unit is weber square per metre or Telsa (T)

**Magnetic dipole moment (M)** – It is the moment of the couple acting on the magnet when placed at right angles to a uniform magnetic field of unit intensity. Its value is:

M = m x 2*I* = 2m*I*

**Magnetic intensity due to a short bar magnet –**

(i) End on position (or Tan-A position)

Where B_{a} = Magnetic intensity at a point distant ‘r’ from the centre and on the axis of the magnet.

(ii) Broad side on position (or Tan-B position)

Where B_{b} = Magnetic intensity at a point distant ‘r’ from the centre of the magnet an on the equatorial line of the magnet.

**Magnetic meridian** – If a magnetic needle is hanged freely from its centre of gravity, then the vertical plane passing through the axis of the magnetic needle, is known as the magnetic meridian.

**Angle of declination** – At any place, the acute angle in between magnetic meridian and the geographical meridian is known as the angle of declination.

**Angle of Dip**– Is the angle at any place which is made in between the horizontal direction and the direction of earth magnetic filed in magnetic meridian. Angle of dip (θ) is –

Where V = Vertical component of Earth’s magnetic field

and H = Horizontal component of Earth’s magnetic field

Torque (τ) acting on a Magnet in a Uniform Magnetic field (B)

τ = 2ml B sin θ = MB sin θ

where m = pole strength of the magnet

21 = effective length of the magnet inclined at angle θ to

and M = 2ml = magnetic dipole moment.

**Pushing Palm Rule for Determining direction of force of Moving charged Particles entering a Uniform Magnetic Filed B in a direction Perpendicular to B–**

Use left palm for negative charges and right palm for positive charges. Keep the palm straight and he thumb outstretched. If fingers point in the direction of B and the outstretched thumb in the direction of the velocity V of the moving charge particle the pushing direction of the palm gives the direction for force F acting charged particle obviously F. B = 0 and F V = 0