A bar magnet of length 1 cm and cross-sectional area 1.0 cm2 produces a magnetic field of 1.5 × 10–4 T at a point in end-on position at a distance 15 cm away from the centre.

(a) Find the magnetic moment M of the magnet.


(b) Find the magnetization I of the magnet.


(c) Find the magnetic field B at the centre of the magnet.


Given:


Length of bar magnet=1cm=10-2m


Cross section area of magnet=1.0cm2 =10-4m2


Magnetic field at a point in end on position =1.5× 10-4T


Distance of point from centre=15cm=15× 10-2m


We need to find the magnetic field at a point in the axis of magnet


Which is given by



Where


M=magnetic moment of the magnet


d=distance of point from centre of magnet


l=half the length of magnet


Proof:


Suppose SN is magnet of length 2l and pole strength m


We need to find the magnetic field at a point P which lies on the axis of magnet at a distance d from the centre.



The magnetic field at P due to north pole of the magnet BN



And it is in rightward direction


Similarly magnetic field at P due to south pole of magnet is given by



Which is in leftward direction(-ve)


The net magnetic field is then given by






Now magnetic moment of magnet is given by


…..(i)


Where


m=pole strength


l=length of magnet


using eqn.(i) we get



Putting the values of l ,d, B and μ0 we get



Solving the equation we get



Therefore magnetic moment M of the magnet is 2.5Am2


Intensity of magnetization(I) is given by formula



Volume of bar magnet =



Where


A=cross-section area of magnet


l=length of bar magnet


hence we get



Putting the values of M, A and l


We get




Therefore magnetization intensity is 2.5× 106A/m


we know that magnetic field at a point P due to a magnetic charge m at a distance d from it is given by



Using eqn.(i) we get


….(ii)


Also magnetizing intensity H is given by formula



Using eqn.(ii) we get



The total magnetic field intensity at the centre of magnet due to magnet is equal to sum of magnetic field intensities due to north pole(HN) and south pole (Hs)



Magnetic field intensity due to north and south pole are equal in magnitude (by symmetry)


∴



Putting the values of M,d and l


We get




Now net magnetic at the centre B is given by



Where


μo=permeability of the free space


H=magnetic field intensity


I=intensity of magnetization


Putting the value of H and I we get




Therefore, magnetic field B at the centre of the magnet is 3.14T


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