Maxwell's Equations: Static or Dynamic Fields?

[Amith2006]

Homework Statement

# Everyone knows that a time varying magnetic field produces an electric field and vice versa. But are the fields produced, static or dynamic? From Maxwell's equations(Faradays and Ampere's law), it seems like they are static. Moreover, Faraday's law of electromagnetic induction doesn't say about production of a time varying electric field due to a change in magnetic flux. Could someone please clarify this concept?

# If a time varying magnetic field can produce an electric field, then it can certainly exert a force on a charge at rest, right?

Homework Equations

The Attempt at a Solution

 

[kuruman]

As I understand the term, "static" means "no explicit time dependence" in the fields. Maxwell's equations become

$$\nabla \cdot D = \rho$$

$$\nabla \cdot B = 0$$

$$\nabla \times E = 0$$

$$\nabla \times H = J$$

These equations have no time-varying fields and the fields are static. If you change your mind and say " I want a time-varying magnetic field to create an electric field", then you are saying that the fields are no longer static. Once you do that, you cannot turn around and ask if they are static because you have already assumed that they are not.

# If a time varying magnetic field can produce an electric field, then it can certainly exert a force on a charge at rest, right?

If by "it" you mean the electric field, the answer is yes. That's how induced currents move around.

 

[Amith2006]

" I want a time-varying magnetic field to create an electric field", then you are saying that the fields are no longer static.

So, a time varying magnetic field produces a static electric field which is evident from
$$\nabla$$ $$\times$$ $$\textit{E}$$ = -dB/dt {Faradays law}
isn't it?

 

[gabbagabbahey]

So, a time varying magnetic field produces a static electric field which is evident from
$$\nabla$$ $$\times$$ $$\textit{E}$$ = -dB/dt {Faradays law}
isn't it?

An electric field produced this way need not be 'static'. In fact, in order for it to be static, you would require the magnetic field to be linear in time (i.e. $B=\alpha t+ \beta$), and in practice, it's impossible to produce such a field since its magnitude would increase without bound as time goes on. You can always produce an induced electric field that is static for a short period of time, but you can never actually induce a truly static electric field in this manner.

 

[Amith2006]

An electric field produced this way need not be 'static'. In fact, in order for it to be static, you would require the magnetic field to be linear in time (i.e. $B=\alpha t+ \beta$), and in practice, it's impossible to produce such a field since its magnitude would increase without bound as time goes on. You can always produce an induced electric field that is static for a short period of time, but you can never actually induce a truly static electric field in this manner.

Thats a more precise answer.Thanx. Suppose u have a coil and magnet. If the magnet is moved with uniform velocity, can we produce a static electric field?

 

[kuruman]

No, because the magnetic field of a magnet is not spatially uniform which means that equal displacements of the magnet do not result in equal changes of the magnetic field.

 

[gabbagabbahey]

The easiest way to produce a quasi-static Faraday field is probably to take two large Helmholtz coils (the B-field in between will be almost uniform) and linearly ramp up the current in the coils during some time interval (you can only do this for a limited time of course before the power needed to continue becomes prohibitively large). If you were to place a small circuit with a voltmeter in between the two coils, you should measure a constant voltage during that time interval (since the induced field is quasi-static, the voltage in the circuit will be aswell).