Biot-Savart Law

Lorentz's Force Equation

Equation

Consider a charge moving in a magnetic field . The force on the moving charge is given as:

If we have both electric field and magnetic field, the total force on the charge is given as:

Cross Product Order

Definition

In Cartesian coordinates:

In Cylindrical Coordinates:

In Spherical Coordinates:

What this means:

The same applies to the rest of the coordinate systems.

Gauss's Law for Magnetism (Second Maxwell Equation)

See Gauss's law

Magnets have a north and south pole. If you break a magnet in half, each piece has its own north and south pole. In the classical world, magnets cannot be monopole. This means the magnetic flux on a closed surface must be 0, and we have our second Maxwell equation:

Equation

(see integral)

Biot-Savart Law due to Current

See current

Consider a circuit with a very thin wire with current placed arbitrarily in space. We want to calculate the magnetic field at point .

We take a small section of the circuit at some arbitrary point , and create a differential length element , whose direction is the direction of the current. Thus the differential current is .
Now, let be this point to .

Figure

The Biot-Savart law is defined as:

Law

where is the electric permeability of free space, where is a tesla.

(see differential)

This equation is similar to the equation for electric field strength:

You can think of acting as the inverse of permittivity of free space in Coulomb's law (where the constant is written ).

Biot-Savart Law for Moving Charges

We know current is charge per unit time, so . Subbing this in:

Law