What exactly is a magnetic field?

Question

If I have a charge $Q$, then at any point in space I can assign an electric field vector to it, $$E(x,y,z)=\frac{KQ}{R^{2}}\hat{l}$$ And it's how one defines an electric field. I was just thinking how can we understand magnetic field in the same sense? Does unit magnetic particles makes sense?

Answer 1

As you have written, the electric field of a charge $Q$ is $$\mathbf{E}(x,y,z)=\frac{1}{4\pi\epsilon_0}\frac{Q}{R^2}\hat{\mathbf{R}}$$

In a similar manner you can write down the magnetic field of a charge $Q$ (see for example here). But here you need also to account for the velocity $\mathbf{v}$ with which the charge is moving. Notice that a cross product is involved, resulting in a magnetic field $\mathbf{B}$ curling around the velocity vector $\mathbf{v}$. $$\mathbf{B}(x,y,z)=\frac{\mu_0}{4\pi}\frac{Q\mathbf{v}}{R^{2}}\times\hat{\mathbf{R}}$$

Actually both formulas above are only approximations because they neglect that the fields progagate not instantaneously but with a finite speed (the speed of light $c$). So they are valid only for slowly moving charges (meaning $v\ll c$).

Answer 2

You wrote in your question that the formula you mentioned is "how one defines an electric field". But that's not really the definition of the electric field; it's a particular formula that applies when you have a stationary source charge and no changing magnetic fields anywhere in the system.

The real definition of the electric field is that it's a vector field such that if a particle with charge $Q$ is placed at a point $r$ then it experiences a force $Q E(r)$. In other words the electric field can be thought of as "force per unit charge". Note that this might not be the only force acting on the particle. The formula you previously gave allows you to calculate the electric field under some conditions; but in other circumstances, a different formula must be used in order to ensure that it gives the value of $E$ that matches up with the empirically observed force per unit charge (not including forces due to fields other than the electric field).

The definition of the magnetic field is that it's a vector field such that if a particle with charge $Q$ is currently at a point $r$ and currently has velocity $v$ then it experiences a force $Qv \times B(r)$.

Answer 3

Electricity and magnetism are described by Maxwell's laws. One of Maxwell's laws is that there are no magnetic monopoles (the equivalent of a single electric charge for magnetic fields). Even if some physical theories postulate the existence of magnetic monopoles we have not found any in nature so far.

The most elementary version of "magnetic charges" are elementary particles with dipole moments (akin to a magnet with two poles). Particles like the electron have an intrinsic magnetic dipole moment. The magnetic field $\mathbf B(\mathbf r)$ created by a magnetic dipole $\mathbf m$ (note it has to be a vector because it depends on where the north pole is pointing at) is roughly given by

$$\mathbf{B}(\mathbf{r})=\frac{\mu_0}{4\pi}\frac{3\mathbf{\hat{r}}(\mathbf{\hat{r}}\cdot \mathbf{m})-\mathbf{m}}{|\mathbf{r}|^3},$$

where $\mathbf r$ is the position vector, $\hat {\mathbf r}=\mathbf r/|\mathbf r|$ and $\mu_0$ is the vacuum permeability. You can understand the magnetism of some materials as the sum of many of these magnetic dipoles.

Note that both the construction of electric fields in terms of charges and the construction of magnetic fields in terms of dipoles is only valid in static configurations. As soon as you consider dynamic problems, moving charges are going to induce magnetic fields and moving dipoles are going to induce electric fields.