Magnetic field

The **magnetic field** is a vector field quantity in electromagnetism describing the strength and direction of the magnetic field.

A magnetic field **B** applies a force **F** on a charge with charge q moving at a velocity **v** equal to

F
→

=
q

v
→

×

B
→

{\displaystyle \vec{F} = q \vec{v} \times \vec{B}}

. If the charge has a nonzero magnetic charge qm, such as in the case where a hypothetical but currently nonphysical magnetic monopole exists, then the force from a magnetic field becomes

F
→

=
q

v
→

×

B
→

q
m

B
→

{\displaystyle \vec{F} = q \vec{v} \times \vec{B} + q\_m \vec{B}}

.

The strength of a magnetic field can be determined using Maxwell's equations. The two equations that define the properties of a magnetic field are Gauss' Law for Magnetism, which gives the divergence of the magnetic field, and Ampére's Law, which gives the curl of the magnetic field given a current density **j** and an electric field **E**. These are:

* ∇
→

⋅

B
→

=
0

{\displaystyle \vec{\nabla} \cdot \vec{B} = 0}
* ∇
→

×

B
→

=

μ
0

ϵ
0

∂

E
→

∂
t

μ
0

j
→

{\displaystyle \vec{\nabla} \times \vec{B} = \mu\_0 \epsilon\_0 \frac{\partial \vec{E}} {\partial t} + \mu\_0 \vec{j}}

When magnetic monopoles are allowed, then Gauss' Law for Magnetism allows for diverging magnetic fields, so for a magnetic charge density ρm it can be modified to become

* ∇
→

⋅

B
→

=

μ
0

ρ
m

{\displaystyle \vec{\nabla} \cdot \vec{B} = \mu\_0 \rho\_m }

From Ampére's law, the Biot-Savart Law can be derived which gives the exact magnitude and direction of the magnetic field **B** at a point **r** near a wire C with length element **l** (with **r'** being the displacement between the length element and **r**) and a current I flowing through it.

B
→

(

r
→

# )

μ
0

4
π

∫
C

I
d

l
→

×

r
′

→

r
′

→

3

{\displaystyle \vec{B} ( \vec{r} ) = \frac{\mu\_0}{4 \pi} \int\_C \frac{I d\vec{l} \times \vec{r'}}{\left|\vec{r'}\right|^3}}

If magnetic monopoles are allowed, then from Gauss' Law, Coloumb's Law for Magnetism can be derived which gives the magnitude and direction of a magnetic field **B** at a point **r** near a point monopole with magnetic charge qm

B
→

(

r
→

# )

μ
0

q
m

r
→

4
π

r
→

3

{\displaystyle \vec{B} ( \vec{r} ) = \frac{\mu\_0 q\_m \vec{r}}{4 \pi \left|\vec{r}\right|^3}}

## Vector Potential

The magnetic field can be found from the curl of a magnetic vector potential

B
→

=
∇
×

A
→

{\displaystyle \vec B = \nabla \times \vec A}

. There is some freedom in picking this potential; any **A** that satisfies this also has an infinite number of corresponding **A** + **∇**φ that also satisfy it (since **∇**φ has a curl of 0). In magnetostatics, a vector potential with zero divergence is typically chosen, in order to simplify calculations. The choice of the vector potential is known as the gauge.

## Scalar Potential

In the absence of of any currents or electric fields, such that the curl of the magnetic field is zero, the magnetic field can also be defined as the gradient of a magnetic scalar potential

B
→

=
∇
Ψ

{\displaystyle \vec{B} = \nabla \Psi}

. This potential is not defined across regions where currents are flowing, and regions separated by surface or volume currents may have completely different scalar potentials defined across them.

There is also some freedom is picking this potential; since the gradient of a scalar is 0, any Ψ that satisfied this has an infinite number of corresponding Ψ+a that also satisfy it. Typically, this is chosen so that Ψ is zero at infinity.

## References

1. ↑ http://en.wikipedia.org/wiki/Magnetic\_field
2. ↑ http://en.wikipedia.org/wiki/Lorentz\_force
3. ↑ 3.0 3.1 3.2 http://en.wikipedia.org/wiki/Magnetic\_monopole
4. ↑ http://en.wikipedia.org/wiki/Gauss%27s\_law\_for\_magnetism
5. ↑ http://en.wikipedia.org/wiki/Amp%C3%A8re%27s\_circuital\_law
6. ↑ http://en.wikipedia.org/wiki/Biot%E2%80%93Savart\_law
7. ↑ https://en.wikipedia.org/wiki/Vector\_potential
8. ↑ https://en.wikipedia.org/wiki/Magnetic\_potential