What is the Biot-Savart Law?

Answer 1

The Biot-Savart Law describes the magnetic fields due to a steady current. It is a simplification of the more general Maxwell equations.

The general equation is described in calculus:

dB = u/(2*pi) * [(J dV) x r]/r_mag^3

dB = differential magnetic field, a vector, measured in (Teslas / meter)

u = permeability of the medium the wire is in. In free space u = uo (4*pi*10^-7 Tesla * meters / Ampere)

J dV = current density vector in a differential volume

'x' = cross product

r = vector to where you want to calculate the magnetic field

r_mag = magnitude of that vector (aka distance)

If you are not familiar with calculus this equation may look rather difficult, but there are a couple of idealizations that simplify this calculation.

1) If our current is only 1 dimensional, the J becomes I (current).

2) By making the wire symmetric (either circular, straight, infinite, etc) we can get a much much much simpler closed form expression.

For an infinitely long, straight piece of wire: B = u*I/(2*p*r) Teslas, r is the distance radially away from the wire.

For a circular loop of wire: B = u*I/(2*R) Teslas, R is the radius of the circular loop.

So enough of the mathmatics, lets talk about what the equations mean. The math says that magnetic fields arise from moving charges (current) and that the magnetic fields are circles around the wire. It also says that it points in a direction (which makes sense since magnetic fields are vector fields). That direction is given by the cross product of the direction the current is going, with the direction away from the wire. The result is a vector that rotates either clockwise or counterclockwise around the wire.

I don't know if your familiar with the cross product, but the common way to picture what happens is to use three fingers, your middle finger, index finger, and your thumb. First make a fist, then point your thumb up, like a thumbs up. Then point your index away, like a pistol, and then make a right angle between your index and middle finger. All three fingers should be right angles with each other. This completely describes the cross product.

The index is the first vector (in this case the current), the middle finger is the second vector (the direction of where you want to calculate the magnetic field), and your thumb is the result (the magnetic field). So a quick example is picture a wire flowing through your monitor. Point your index finger in this direction. Now you want to calculate the direction of the magnetic field at the top left corner of your screen. So make your middle finger point to the top left corner. The direction your thumb is point in is the direction of the magnetic field (it should be pointing at the top right corner).

So I hope explained this well enough so you can calculate the field in some situations and can picture what the field looks like, all described by the Biot-Savart Law.