Chapter 30: Sources of the Magnetic Field

• Biot-Savart law: The magnetic field contribution from a line element ds of wire carrying a current I is

dB = (μ₀/4π) (I ds × r)/r² ,  where  μ₀ = 4π × 10^–7 T·m/A  and  r  is a unit vector .

• Long line of current: The magnetic field due to a long straight wire carrying a current i is

B = (μ₀/2π) I/r . 

• Arc of wire: The magnetic field due to a circular wire carrying a current I and forming an angle φ is

B = (μ₀/4π) I φ/r .

• Magnetic force between wires carrying currents: For two long straight wires a distance a apart,

F = (μ₀/2π) (I₁I₂/a) L .

As can be seen from the right-hand rule, the force is attractive for currents in the same direction, and repulsive for forces in opposite directions.

• Ampère's law: For any closed loop C in space enclosing a current I_enc,

∫_C B · ds = μ₀ I_enc .
 

• Magnetic field of a long solenoid: The magnitude of the field is

B = μ₀nI      (n = N/L) ,

and the direction is obtained from the right-hand rule. [We did not cover the magnetic field of a toroid.]

• [Magnetic behavior of materials: Diamagnetism, paramagnetism and ferromagnetism.]