JEE Physics Magnetism and Matter Notes Set 02

Magnetism and Matter: Quick Revision

A bar magnet behaves like a current loop. Its magnetic moment is \( M=m\times2\ell \) (pole strength times length). On the axis the field is \( \frac{\mu_0}{4\pi}\frac{2M}{r^3} \) and on the equatorial line \( \frac{\mu_0}{4\pi}\frac{M}{r^3} \), so the axial field is twice the equatorial one at the same distance. At angle \( \theta \) from the axis, \( B=\frac{\mu_0}{4\pi}\frac{M\sqrt{1+3\cos^2\theta}}{r^3} \). In a uniform field the torque is \( \vec M\times\vec B \), the energy is \( -\vec M\cdot\vec B \), and the net force is zero.

Earth's Magnetism

The vertical plane through the magnetic poles is the magnetic meridian. The angle between the earth's total field and the horizontal is the angle of dip \( \delta \); the angle between the magnetic and geographic meridians is the declination. The horizontal component is \( B_H=B\cos\delta \) and the vertical component is \( B_V=B\sin\delta \), so \( \tan\delta=\frac{B_V}{B_H} \). Dip is 0 degrees at the magnetic equator and 90 degrees at the poles. In two vertical planes at right angles, the apparent dips satisfy \( \cot^2\delta=\cot^2\delta_1+\cot^2\delta_2 \). A vibration magnetometer measures field through \( T=2\pi\sqrt{\frac{I}{MB}} \), so the frequency goes as \( \sqrt B \).

Magnetic Properties of Materials

The key relations are \( B=\mu H=\mu_0(H+I) \), \( \mu_r=1+\chi \) and \( \chi=\frac IH \). Above the Curie temperature a ferromagnet turns paramagnetic (Curie-Weiss law, \( \chi\propto\frac1{T-T_C} \)). Superconductors are perfect diamagnets.

Hysteresis and Uses

The B-H loop shows retentivity (field left when H is zero) and coercivity (reverse field needed to demagnetise). Materials for permanent magnets, such as steel and alloys of cobalt, need high retentivity and high coercivity, giving a wide loop. Soft iron has low retentivity and low coercivity with a narrow loop, so it suits electromagnets, transformer cores and magnetic shielding. The area of the loop equals the energy lost per cycle per unit volume.