# Moving charges and magnetic field

## Homework Statement

Why does a moving charge act as a source of magnetic field?

## The Attempt at a Solution

I have to idea as to how I should proceed!

## Answers and Replies

There is no simple explanation. You will have to look into quantum electrodynamics.

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Would the answer be "because there is a changing electric field"?

An explanation can be found in classical EM theory just as well. It has to do with how the EM field transforms from one inertial reference frame to another. The derivation is carried out quite beautifully in the Berkley Physics Course book on EM.
I suggest you see if your library has it, or if you can access it through a friend or some other means.

McGraw-Hill Book Company - Berkley Physics Course - Volume 2 - Electricity and Magnetism
by Edward M. Purcell
ISBN - 07-004859-2 (For my edition)

There is no simple explanation. You will have to look into quantum electrodynamics.

Why is that so?
And what is the explanation anyway?

Would the answer be "because there is a changing electric field"?

Why?

ehild
Homework Helper
It is experimental fact that magnetic field can be observed around current carrying wires by a simple compass. Electric current is flow of charged particles. Flow is some arranged motion of those particles.

ehild

Why?

As ehild said, people did/do experiments with moving charges and they can detect a magnetic field, once the charges stop moving the magnetic field disappears (in their frame). These same people can create formulas that explain the experiments nicely, but they probably cant tell you "why" it happens. My book used the Biot-Savart law http://en.wikipedia.org/wiki/Biot–Savart_law when they introduced the concept.

A more thorough "why" other than quoting experiments is probably what QED is trying to (or already has) accomplished. If you're in a lower level physics course that asked the question then they are probably just looking for the answer of "because there is a changing electric field". This is more accurate than saying the "because a charge is moving (changing)", because you'll learn later that an electromagnetic field can exist independent of charges; a changing electric field induces a magnetic field which in turn induces an electric field, all without a charge present.

"Why does a moving charge act as a source of magnetic field?" Think of it like this, a moving charge isn't the source of the magnetic field. A charge is the source of an electric field, moving this charge causes this electric field to move, or "change", and it is this changing electric field alone that induces a magnetic field.

HallsofIvy
Science Advisor
Homework Helper
You don't need "quantum electrodynamics", just "Maxwell's equations" or, even more specifically, the "Ampere's circuit law" equation ( with no current), one of the four "Maxwell's equations" that govern the relationship between electric and magnetic fields (being, together, the "electro-magnetic field"):

$$\nabla\times \vec{B}= \mu_0\nu_0\frac{\partial \vec{E}}{\partial t}$$
Where $\vec{B}$ is the magnetic vector field, $\vec{E}$ is the electric vector field, $\mu_0$ is the "permeability of free space", and $\nu_0$ is the "permittivity of free space", both universal constants.

When a charge is static, it has a constant electric field and so its time derivative is 0- 0 magnetic field. When a charge is moving, its electric field moves with it so it has a changing electric field. The derivative is non-zero so the magnetic field is non-zero.

Suppose I have a long current-carrying wire, we know that there is a magnetic field around the wire. But, there is no electric field around the wire (because, the wire is neutral).

How would Maxwell's equations explain this?

ehild
Homework Helper
The Maxwell equation in vacuum HallsofIvy cited is in its full form :

$$\nabla\times \vec{B}= \mu_0 \epsilon_0\frac{\partial \vec{E}}{\partial t}+\mu_0 \vec j$$

where E is electric field, μ0 is the magnetic permeability of the free space, ε0 is the permittivity of free space, and j is the vector of the external electric current density. So this equation includes the magnetic field caused both by the time- dependent electric field and the external currents.

ehild

Yes, but the question was why, "why does a current produce a magnetic field" ?

Yes, but the question was why, "why does a current produce a magnetic field" ?

Because charges have an E-field associated with them and special relativity works. The derivation is fairly long, but the gist of it is that if you allow the E and B fields to be local properties, and you accept the postulates of relativity, carrying out the appropriate Lorentz Transformations tells you how the E and B fields transform.

The full transform, is as follows, from page 213 in the book I previously cited:

We give below the full list of transformations. All primed quantities are measured in the frame F', which is moving the positive x direction with speed v as seen from F. Unprimed quantities are the numbers which are the results of measurements in F. As usual, $$\beta$$ stands for $$\tfrac{v}{c}$$ and $$\gamma$$ for $$\frac{1}{\sqrt{1-\beta ^2}}$$

$$E'_x=E_x$$
$$E'_y=\gamma (E_y - \beta B_z)$$
$$E'_z=\gamma (E_z + \beta B_y)$$

$$B'_x=B_x$$
$$B'_y=\gamma (B_y + \beta E_z)$$
$$B'_z=\gamma (B_z - \beta E_y)$$

Note that these transformations are given in Gaussian CGS units, for the SI variety, just multiply and divide by c where necessary so that the units fit.

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ehild
Homework Helper
"why does a current produce a magnetic field" ?

What else could product magnetic field?

Why does a charge produce electric field? Why do objects with mass attract each other? It is the nature of the magnetic field that it exist around any kind of electric current. It can be conduction current flowing in a wire or displacement current caused by moving charges in the atoms or molecules, and even a changing electric field. We do not know why does it exist but we can observe it. The magnetism in materials is accounted for the angular momentum of the electrons (moving charges, again) and for spin, kind of angular momentum again.

ehild