## Magnetic Field Intensity Due To A Magnetic Dipole (Bar Magnet) Along Its Axis And Perpendicular To Its Axis

• The magnetic field intensity due to a bar magnet at any point on the axial line is B10/4π)(2md/(d2-l2)2) where d is the distance of the point from the centre of the magnet.
• Magnetic field intensity due to a bar magnet at any point on the equatorial line of the bar magnet is
B2=  (µ0/4π)(2md/(d2+l2)3/2)
• Direction of B1 is along SN
• Direction of B2 is along a line parallel to NS.

## Torque On A Magnetic Dipole (Bar Magnet) In A Uniform Magnetic Field

$$\text{When a magnetic dipole of dipole moment}\space \vec{\text{M}} \text{is placed in a uniform magnetic field}\space \vec{\text{B}},\\ \text{it will experience a torque and is given by}\space \vec{\text{τ}}=\vec{\text{M}}×\vec{\text{B}} \space\text{or}\space\vec{\text{τ}}=\text{MB\space sin}\space\theta, \\ \text{where}\space\theta\space \text{is the angle between} \space\vec{\text{M}} \space\text{and} \space\vec{\text{B}}$$

## Bar Magnet As An Equivalent Solenoid

• A bar magnet and a solenoid (for a far axial point) produces similar magnetic fields.

B =(µ0/4π)(2M/r3),

where M is the magnetic moment of the solenoid and is given by, M = n (2l) × I × (πa2)

n = number of turns per unit length of solenoid

2l = length of solenoid

I = current passed through the solenoid

r = distance between the point and centre of the solenoid

## Magnetic Field Lines

• Magnetic field line is an imaginary curve, the tangent to which at any point gives the direction of magnetic field $$\vec{\text{B}}\space \text{at that point.}$$
• Larger the number of field lines crossing per unit area, stronger is the magnitude of magnetic field.
• Magnetic field lines do not intersect.

## Para-dia And Ferro-magnetic Substances With Examples

• Paramagnetic substances are those, which when placed in a magnetic field are feebly magnetised in the direction of the magnetising field. Aluminium, platinum, and chromium are some examples of paramagnetic substances.
• Paramagnetic substances has small and positive value of magnetic susceptibility (Χm).
• Diamagnetic substances are those which when placed in a magnetising field are feebly magnetised in a direction opposite to that of the magnetising field. Copper, zinc, bismuth, gold, silver, etc., are some of the examples of diamagnetic substances.
• Ferromagnetic substances are those, which when placed in a magnetic field are strongly magnetised in the direction of magnetising field. Iron, nickel and cobalt are the examples of ferromagnetic substances.
• Diamagnetic substances has a small and negative value of magnetic susceptibility (χ m).
• Ferromagnetic substances has large and positive value of magnetic susceptibility (χm).

## Electromagnets And Factors Affecting Their Strengths

• Electromagnets are made of ferromagnetic materials which have high permeability and low retentivity, such as soft iron.
• Factors affecting the strength of electromagnets are :

(i) Number of loops in the coil

(ii) Strength of current passed

(iii) Nature of core of the electromagnet

(iv) Temperature

• Electomagnets are used in electric bells, loudspeakers and telephone diaphragms.
• Giant electromagnets are used in cranes to lift machinery and heavy loads.