JEE Physics Magnetism and Matter Notes Set 01

Bar Magnet and Magnetic Dipole

A bar magnet behaves as a magnetic dipole of moment M (direction S to N). Field lines are closed loops that never intersect.
On the axis at distance r (r much greater than the magnet length): \( B=\frac{\mu_0}{4\pi}\frac{2M}{r^3} \). On the equatorial line: \( B=\frac{\mu_0}{4\pi}\frac{M}{r^3} \), in the direction opposite to M. The axial field is twice the equatorial field at equal distance.
In a uniform field B a dipole feels torque \( \tau=MB\sin\theta \), potential energy \( U=-MB\cos\theta \), and no net force. Work needed to rotate it from angle \( \theta_1 \) to \( \theta_2 \) is \( MB(\cos\theta_1-\cos\theta_2) \).
Small oscillations about the field direction have period \( T=2\pi\sqrt{\frac{I}{MB}} \).

Earth's Magnetism

The three elements at a place are the declination (angle between magnetic and geographic meridians), the dip \( \delta \) (angle the total field makes with the horizontal), and the horizontal component \( B_H \). They are related by \( B_H=B\cos\delta \), \( B_V=B\sin\delta \), \( \tan\delta=B_V/B_H \). Dip is 0 at the magnetic equator and 90 degrees at the magnetic poles.
Tangent galvanometer: \( B_{coil}=B_H\tan\theta \), so \( I=\frac{2RB_H}{\mu_0N}\tan\theta \).
Vibration magnetometer: \( T=2\pi\sqrt{I/MB_H} \), so the frequency varies as the square root of the field. A neutral point is a point where the field of the magnet cancels the earth's horizontal field.

Gauss's Law for Magnetism and Magnetisation

Isolated magnetic poles do not exist, so the net magnetic flux through any closed surface is zero: \( \oint\vec B\cdot d\vec A=0 \).
Magnetisation M is the dipole moment per unit volume. Magnetic intensity H and field B are connected by \( \vec B=\mu_0(\vec H+\vec M) \). For linear materials \( M=\chi H \), \( \mu_r=1+\chi \) and \( B=\mu_0\mu_rH \).

Types of Magnetic Materials

Diamagnetic: small negative \( \chi \), \( \mu_r<1 \), weakly repelled, independent of temperature, no permanent atomic moment (copper, bismuth, water). A rod aligns across the field.
Paramagnetic: small positive \( \chi \), \( \mu_r \) slightly above 1, weakly attracted. Curie's law \( \chi=C/T \) (aluminium, sodium, oxygen). A rod aligns along the field.
Ferromagnetic: large positive \( \chi \), strongly attracted, domains, hysteresis. Above the Curie temperature the substance becomes paramagnetic and follows \( \chi=\frac{C}{T-T_c} \) (iron, cobalt, nickel).

Hysteresis and Uses

The B-H loop of a ferromagnet is a closed curve whose area is the energy lost per cycle per unit volume. Retentivity is the leftover B at H = 0 and coercivity is the reverse H needed to bring B to zero.
Soft iron: narrow loop, low coercivity and low retentivity, so it suits transformer cores, electromagnets and magnetic shielding. Steel and alloys such as alnico: broad loop, high coercivity and retentivity, so they suit permanent magnets.

Solved Examples

Illustration 1. A bar magnet of moment 2 A m² is placed in a field of 0.5 T at 30 degrees to the field. Find the torque and the work needed to turn it to the direction of the field.
Answer:
\( \tau=MB\sin30^\circ=2\times0.5\times0.5=0.5 \) N m.
Work done by the field in turning it to alignment: \( W=MB(\cos0^\circ-\cos30^\circ)=2\times0.5\times(1-0.866)=0.134 \) J.
Torque = 0.5 N m; the field does 0.134 J of work in aligning it.

Illustration 2. At a place the horizontal component of the earth's field is 0.3 gauss and the dip is 60 degrees. Find the total field.
Answer:
\( B=\frac{B_H}{\cos\delta}=\frac{0.3}{\cos60^\circ}=0.6 \) gauss.
B = 0.6 gauss (6 × 10⁻⁵ T).

Illustration 3. A magnet oscillates 20 times per minute in a field of 0.4 gauss. At another place it oscillates 30 times per minute. Find that field.
Answer:
\( B\propto f^2 \): \( B_2=0.4\times\left(\frac{30}{20}\right)^2=0.9 \) gauss.
B = 0.9 gauss.

Illustration 4. A paramagnetic sample has susceptibility 0.003 at 300 K. Find its susceptibility at 200 K.
Answer:
\( \chi\propto1/T \): \( \chi_2=0.003\times\frac{300}{200}=0.0045 \).
0.0045.