Amperes law and electrodynamics and Gauss's law

[Maxwellkid]

It's evident that ampere's law is used to calculate the magnetic field produced by flowing charges. Can I use Gauss's Law to calculate the electric field produced by the flowing charges also?

I ask this question because in the middle of a solid conductor, there is a flow of charge. However, in the static case there is no charge in the middle of a solid conductor. So, does Gauss's law inside a solid conductor hold as long as I know how much charge is enclosed at a given time?

 

[dx]

So, does Gauss's law inside a solid conductor hold as long as I know how much charge is enclosed at a given time?

Gauss' law, being one of Maxwell's equations, always holds, whether you know how much charge is enclosed or not.

 

[Maxwellkid]

Gauss' law, being one of Maxwell's equations, always holds, whether you know how much charge is enclosed or not.

however, isn't it true that the electric field inside a solid conductor is zero?

 

[Doc Al]

however, isn't it true that the electric field inside a solid conductor is zero?

Not if charge is flowing. (The electric field is zero within conductors in electrostatic equilibrium.)

 

[I_am_learning]

Gauss law Gives the electric Filed Setup by Stationary Charges Only. And it applies at every case.
Flowing Charges don't produce extra electric Fields, they in fact produce magnetic Filed which is calculated from Amperes law.
There are other laws about these.
Faraday's Laws Gives the Electric Field Produced By changing magnetic Field and Modified Amperes' law gives magnetic Field produced by changing Electric Field.
Moreover like the gauss law in electrostatics there is gauss law for magnetostatics that gives the magnetic Field produced by stationary magnetic charges (poles).

 

[Born2bwire]

Not if charge is flowing. (The electric field is zero within conductors in electrostatic equilibrium.)

In a perfect conductor, the electric field is always zero, regardless of statics. However, the movement of charges via currents is contained completely on the surface of the perfect electrical conductor.

 

[Doc Al]

In a perfect conductor, the electric field is always zero, regardless of statics. However, the movement of charges via currents is contained completely on the surface of the perfect electrical conductor.

Presumably the OP is talking about a typical non-perfect conductor, such as copper, within which there can be a non-zero field and a current.

 

[Maxwellkid]

Gauss law Gives the electric Filed Setup by Stationary Charges Only. And it applies at every case.
Flowing Charges don't produce extra electric Fields, they in fact produce magnetic Filed which is calculated from Amperes law.
There are other laws about these.
Faraday's Laws Gives the Electric Field Produced By changing magnetic Field and Modified Amperes' law gives magnetic Field produced by changing Electric Field.
Moreover like the gauss law in electrostatics there is gauss law for magnetostatics that gives the magnetic Field produced by stationary magnetic charges (poles).

is that statement true? flowing charges don't produce extra electric fields?

 

[maverick280857]

Gauss law Gives the electric Filed Setup by Stationary Charges Only. And it applies at every case.

thecritic, what do you mean by "only" here? To be sure, if you mean that Gauss's law does not apply to non-stationary charges, you are wrong.

To quote Purcell (Berkeley Physics Course Volume 2, Second Edition, page 175):

We define the amount of charge inside $S$ as $1/4\pi$ times this integral:

$$Q = \frac{1}{4\pi}\int_{S(t)}\mathbf{E}\cdot d\mathbf{a}$$

It would be embarrassing if the value of $Q$ so determined depended on the size and the shape of the surface $S$. For a stationary charge it doesn't -- that is Gauss's Law. But how do we know that Gauss's law holds when charges are moving? Fortunately it does. We can take that as an experimental fact. This fundamental property of the electric field of moving charges permits us to define the quantity of charge by Eq. 3 (ed - the above equation). From now on we can speak of the amount of charge in a region or on a particle, and that will have a perfectly definite meaning even if the charge is in motion.

I hope that clears the confusion. [ED - Read $1/\epsilon_0$ instead of $1/4\pi$ if you use SI units. Anyway, the physics remains the same.]

is that statement true? flowing charges don't produce extra electric fields?

What do you mean by "extra"? The total electric field and magnetic field in a region, produced by a charge in motion (or at rest), are solutions to Maxwell's equations, of which Gauss's Law is one.

 

[I_am_learning]

thecritic, what do you mean by "only" here?

Opps, I realize now that "only" wasn't necessary. I am sorry about its misdirection.

In fact Gauss Law is always applicable. Since it gives the electric field based on amount of charge enclosed at any instant, we can say it gives instanteneous Electric Field, which is no different than the persistent electric field if uniform current of charge is flowing.

What do you mean by "extra"?

I simply wanted to say that, given a charge at rest at certain distance and given another charge ,which is in motion, but presently is at the same distance; then the electric Fields by both of these charges are equal.
Thats why I said moving charges don't produce EXTRA (more than that it would have produced were it at rest) electric Field.

 

[maverick280857]

I simply wanted to say that, given a charge at rest at certain distance and given another charge ,which is in motion, but presently is at the same distance; then the electric Fields by both of these charges are equal.
Thats why I said moving charges don't produce EXTRA (more than that it would have produced were it at rest) electric Field.

The extra question was directed to Maxwellkid, but that's okay :-)

For OP and future visitors to this thread:

Anyway, for the record (in order not to confuse OPs or beginning students of EM theory who might stumble upon this thread), for a good treatment of the fields produced by a charge in motion, refer to Purcell's book (details in my previous post). The pictures in his book are very helpful in understanding the nature of the field patterns. And I would like to state here (for beginning students) that this is a nontrivial subject and so the interpretation of the equations must be made clear, and so must the inputs from experiments be distinguished from the fundamental assumptions made while deriving these expressions.

The connection between field transformations and the Lorentz force law must be understood properly. Secondly, ALL classical electrodynamic phenomena can be explained using Maxwell's equations and the Lorentz Force law. So, these equations (of which Gauss's Law is one) are valid in all inertial frames of reference.