Magnetism and magnetic effects of electric current

Baxial = (µ0/4π) × 2pm / r³

Step 1: For a short bar magnet (r >> l), the magnetic field at a point on the axial line is derived using contributions from both north and south poles. Step 2: After simplification for r >> l, the formula is Baxial = (µ0/4π) × (2pm/r³). Step 3: Note the factor of 2 in the numerator — this is important. Step 4: On the equatorial line, the field is Bequatorial = (µ0/4π) × pm/r³ (no factor of 2). Step 5: So the axial field is exactly twice the equatorial field in magnitude.

τ = pm × B × sin θ

Step 1: A bar magnet in a uniform magnetic field experiences two equal and opposite forces on its poles. Step 2: These forces form a couple (torque) tending to align the magnet along the field. Step 3: The magnitude of torque is τ = pm B sin θ, where θ is the angle between pm (dipole moment) and B. Step 4: In vector form, τ⃗ = p⃗m × B⃗. Step 5: When θ = 90°, torque is maximum (τ = pmB); when θ = 0° or 180°, torque is zero. The cos θ option would give maximum torque at θ = 0° which is physically wrong.

Diamagnetic material

Step 1: Magnetic susceptibility χm = M/H gives the response of a material to an applied magnetic field. Step 2: Diamagnetic materials develop magnetisation opposite to the applied field, so M is negative, giving negative χm. Step 3: Paramagnetic materials have small positive susceptibility (they weakly align with the field). Step 4: Ferromagnetic materials have large positive susceptibility (they strongly align with the field). Step 5: Examples of diamagnetic materials: Copper, Bismuth, Gold, Water.

90°

Step 1: Biot-Savart's law states dB = (µ0/4π) × (Idl sin θ)/r². Step 2: The magnetic field depends on sin θ, where θ is the angle between the current element Idl and the position vector r̂. Step 3: sin θ is maximum when θ = 90°, giving maximum dB. Step 4: When θ = 0° or 180° (point lies on the conductor), sin θ = 0, so dB = 0 (minimum). Step 5: This is why the magnetic field is zero exactly along the direction of current flow.