JEE Physics Magnetism and Matter Notes Set 04

Important Notes and Formulas

Magnet and Magnetic Dipole

Every magnet has a north and a south pole, like poles repel and unlike poles attract, and a freely suspended magnet comes to rest along the north-south direction. Isolated magnetic poles (monopoles) do not exist. The effective magnetic length of a bar magnet is about 0.84 times its geometric length.
Magnetic dipole moment of a bar magnet of pole strength \( m \) and length \( 2l \): \( M=m\cdot2l \) (unit A m\( ^2 \)); it points from the south pole to the north pole.
A current loop behaves as a magnet: \( M=NiA \). For an electron of charge \( e \) moving with speed \( v \) in a circle of radius \( r \), \( i=\frac{e}{T}=\frac{ev}{2\pi r} \) and \( M=iA=\frac{evr}{2} \).

Field of a Short Bar Magnet

On the axial line at distance \( r \) from the centre: \( B=\frac{\mu_0}{4\pi}\cdot\frac{2M}{r^3} \).
On the equatorial line: \( B=\frac{\mu_0}{4\pi}\cdot\frac{M}{r^3} \). At the same distance the axial field is twice the equatorial field, and the two fields point in opposite directions relative to the dipole moment.

Dipole in a Uniform Field

The two poles feel equal and opposite forces, so the net force is zero but there is a torque: \( \tau=MB\sin\theta \), that is \( \vec\tau=\vec M\times\vec B \). The torque is largest (\( MB \)) at \( \theta=90^\circ \) and zero at \( \theta=0^\circ \).
Work done in turning the dipole from \( \theta_1 \) to \( \theta_2 \): \( W=MB(\cos\theta_1-\cos\theta_2) \).

Example 1. A short bar magnet of dipole moment \( 2\ \text{A m}^2 \) is placed in air. Find the field on its axis at 0.2 m from the centre and on the equatorial line at the same distance.
Solution:
Axial: \( B=10^{-7}\times\frac{2\times2}{(0.2)^3}=10^{-7}\times500=5\times10^{-5} \) T.
Equatorial: \( B=10^{-7}\times\frac{2}{(0.2)^3}=2.5\times10^{-5} \) T, half of the axial value.

Earth's Magnetic Field

The earth behaves like a huge magnet whose magnetic poles are not at the geographic poles. Three elements describe the field at a place. The angle of declination is the angle between the geographic meridian and the magnetic meridian. The angle of dip \( \delta \) is the angle that a freely suspended magnet (dip needle) makes with the horizontal in the magnetic meridian. The horizontal component \( B_H \) is the component of the earth's field along the horizontal.
\( B_H=B\cos\delta \), \( B_V=B\sin\delta \), \( \tan\delta=\frac{B_V}{B_H} \) and \( B=\sqrt{B_H^2+B_V^2} \).

Example 2. At a place the horizontal component of the earth's field is 0.3 G and the angle of dip is \( 30^\circ \). Find the vertical component and the total field.
Solution:
\( B_V=B_H\tan30^\circ=0.3\times0.577=0.17 \) G.
\( B=\frac{B_H}{\cos30^\circ}=\frac{0.3}{0.866}=0.35 \) G.

Magnetic Materials

Paramagnetic substances have atoms with an odd number of electrons in the outermost orbit, so each atom has a net dipole moment. They get weakly magnetised along an applied field, the field lines are drawn into the substance, and their susceptibility increases as the temperature falls (Curie's law).
Diamagnetic substances have all electrons paired, so atoms have no net dipole moment. In an applied field they are weakly magnetised in the opposite direction, the field lines are pushed out of the substance, and the effect is almost independent of temperature.
Ferromagnetic substances have atoms with some unpaired electrons and form domains of aligned dipoles. A magnetising field makes the domains line up and gives a very strong magnetisation. Above the Curie temperature a ferromagnetic substance turns paramagnetic.

Hysteresis and Practical Magnets

The graph of magnetic induction \( B \) against magnetising field \( H \) over a full magnetising cycle is the hysteresis (B-H) loop. The work done (energy lost as heat) in one complete cycle of magnetisation and demagnetisation equals the area of the loop. Permanent magnets are made of steel, which keeps its magnetism.
An electromagnet is a soft iron core inside a solenoid carrying current; it stays magnetised only while the current flows. Its strength increases with more turns, more current and a soft iron core.