Electromagnetism/Reference/Ampere's Law

Ampere's circuital law

"Ampere's circuital law" (named after André-Marie Ampère, not directly named after the unit of current), gives the magnetic field in the vicinity of an infinitely long straight wire carrying an electric current.

The magnetic field goes in circles around the wire, following a right-hand rule and calculated by 
$B={\frac {\mu \ I}{2\pi R}}=LI$

With

$B$ . Magnetic field strength

$I$ . Current

$\mu$ . Permeability

$R$ . Radius

$L$ . Inductance

$L={\frac {\mu \ }{2\pi R}}$

Derivation

The pure (almost uselessly so) formula for the magnetic force is that the force acting on moving particle 2 from the motion of particle 1 is:

$F={\frac {\mu \ q_{1}\ q_{2}}{4\pi r^{2}}}\ {\vec {V_{2}}}\times ({\vec {V_{1}}}\times {\hat {r}})$

where $q_{1}\,$ and $q_{2}\,$ are the charges on the particles, ${\vec {V_{1}}}\,$ and ${\vec {V_{2}}}\,$ are their velocities, and ${\hat {r}}\,$ is the unit vector giving the direction from particle 1 to particle 2.

The constant defining the strength of the magnetic force is ${\frac {\mu }{4\pi }}\,$. The value of μ is (by definition! See below) $4\pi \times 10^{-7}$ newtons per ampere squared, or henries per meter.

When this is reformulated in terms of the magnetic field (called B), the force felt by charge q₂ moving at velocity v₂ is

${\vec {F}}=q_{2}\ {\vec {v_{2}}}\times {\vec {B}}\,$

and the magnetic field generated by charge q₁ moving at velocity v₁ is

${\vec {B}}={\frac {\mu \ q_{1}}{4\pi r^{2}}}\ {\vec {V_{1}}}\times {\hat {r}}$

In practical electrical circuits, one uses a continuous flow of current through a wire rather than a static charge moving at a certain velocity. We use the equivalence:

$q\ {\vec {v}}=I\ {\vec {l}}\,$

where I is the current and ${\vec {l}}\,$ is the length and direction of the wire through which the current is flowing. To see this equivalence, observe that a current of I amperes over a wire of length l meters will send I coulombs of charge a distance l every second, equivalent to I coulombs traveling at a speed of l meters per second.

Now we can't have current just flowing through a short length of wire—it has to complete a circuit. So we have to make the wire infinitely long, or else arrange for the current return path to be far away from the region of interest, and integrate the contributions to the magnetic field along the length of the wire that we are interested in.

The magnetic field near a wire carrying current.

So, in terms of wires and current, the magnetic force is

${\vec {F}}=I\ {\vec {l}}\times {\vec {B}}\,$

and the magnetic field (this is the actual law of Biot and Savart) is

${\vec {B}}={\frac {\mu \ I}{4\pi r^{2}}}\ {\vec {l}}\times {\hat {r}}$

In the diagram at the right, we have an infinitely long straight wire carrying current I, and we wish to find the magnetic field at some point at a distance R from the wire. We first calculate the contribution to that field from a small section of the wire, of length dl. That contribution is:

${\vec {B}}={\frac {\mu \ I}{4\pi r^{2}}}\ {\vec {dl}}\times {\hat {r}}$

The cross product shows that the magnetic field vector points directly out of the page/screen, and has magnitude:

$B={\frac {\mu \ I}{4\pi r^{2}}}\ \sin \theta \ dl={\frac {\mu \ I}{4\pi r^{3}}}\ R\ dl$

To get the magnetic field, at that point, from the current over the entire wire, integrate:

$B={\frac {\mu \ I\ R}{4\pi }}\int _{-\infty }^{\infty }{\frac {dl}{(R^{2}+l^{2})^{3/2}}}={\frac {\mu \ I\ R}{4\pi }}{\frac {1}{R^{2}}}\left.{\frac {l}{\sqrt {R^{2}+l^{2}}}}\right|_{-\infty }^{\infty }$

Which is:

$B={\frac {\mu \ I}{2\pi R}}=LI$

With

$B$ . Magnetic field strength

$I$ . Current

$\mu$ .

$R$ . Radius

$L$ . Inductance

$L={\frac {\mu \ }{2\pi R}}$

How is μ Defined?

The quantity $\mu \,$, known as the "permeability of the vacuum", is a physical constant that defines the strength of the magnetic and electric fields. Suppose we have two infinitely long straight parallel wires, separated by distance R, and carrying currents of I₁ and I₂. The magnetic field created by the first wire, at the location of the second wire, is:

$B={\frac {\mu \ I_{1}}{2\pi R}}$

The force on the second wire, along a section of length l, is:

${\vec {F}}=I_{2}\ {\vec {l}}\times {\vec {B}}\,$

If the currents are flowing in the same direction, the wires will be attracted to each other with a force per unit length of:

${\vec {F}}={\frac {\mu \ I_{1}I_{2}}{2\pi R}}\,$

The ampere is defined by means of this formula. An ampere is the amount of current, flowing in each of two parallel wires separated by a distance of one meter, that causes an attractive magnetic force of $2\times 10^{-7}$ newtons per meter of the wires' length. This has the effect of defining

$\mu =4\pi \times 10^{-7}\,$

in SI units, which are newtons per ampere squared, or henries per meter.