- The Biot-Savart law asserts that the magnetic field dB due to an element dl carrying a steady current i at a point P at a distance r from the current element is:
- The constant m0 has the exact value
m0 = 4p x 10-7 T m A-1It is the permeability of free space. It is related to e0, the permittivity of free space and the speed of light in free
space c, by 
- he magnitude of the magnetic field at a distance R from a long, straight wire carrying a current i is given by:
The field lines are circles concentric with the wire.
- The magnitude of the magnetic field due to a N-turn circular coil of radius R carrying a current i at an axial distance x from the center is
At the center this reduces to 
- Ampere's Circuital Law: We have discussed a simplified form of this law. If B is directed along the tangent to every point on the perimeter L of a closed curve and is constant in magnetic along perimeter then.
BL = m0 iewhere ie is the net, current enclosed by the closed circuit.
- Employing Ampere's law one can show that the magnitude of the field B inside a long solenoid carrying a current i is
B = m0 ni
where n is the number of turns per unit length. For a toroid one obtains,
where N is the total number of turns and r the average radius.
- The total force on a charge q moving with velocity v in the presence of magnetic and electric fields B and E, respectively is called the Lorentz force. It is given by the expression:
The magnetic force q (v x B) is normal to v and work done by it is zero.
- In a uniform magnetic field B, the charge q described in (7) executes circular orbit in the plane normal to B. Its frequency of uniform circular motion is called the cyclotron frequency and is given by:
This frequency is independent of the particle's speed and radius. This fact is exploited in a machine, the cyclotron, which is used to accelerate charged particles.
- A straight conductor of length l and carrying steady current i experiences a force F in a uniform external magnetic field B,
F = il x B
where |l| = l and the direction of l is given by the direction of the current.
- Parallel currents attract and anti-parallel currents repel.
- A planar loop carrying a current i, having N closely, wound turns, and an area A possesses a magnetic moment m where, m = N i A and the and the direction of m is given by the right-hand thumb rule: curl the palm of your right hand along the loop with the fingers pointing in the direction of the current. The thumb sticking out gives the direction of m (and A).
When this loop is placed in a uniform magnetic field B, the force F on it isF = 0
And the torque on it is,
t = m x B
In a moving coil galvanometer, this torque is balanced by a counter - torque due to a spring, yielding.
kf = Ni AB
where f is the equilibrium deflection and k the torsion constant of the spring.
- An electron moving around the central nucleus has a magnetic moment ml given by:
where l is the magnitude of the angular momentum of the circulating electron about the central nucleus. The
smallest value of ml is called the Bohr magneton mB and it is
mB = 9.27 x 10-24J/T
