What is the effect of time-varying current on Gauss law in electrodynamics?

In summary, a wire can carry a current and still have a zero net charge distribution throughout, as the number of protons and electrons remain equal regardless of the current magnitude. This only applies to a current varying in time, not in space.
  • #1
quietrain
655
2
if i have a current carrying straight long wire, I = I0sin(wt)

why is gauss law ∇.E = 0?

i thought only for steady currents , then the charges reside on surface, that's why 0 charge enclosed, and hence gauss law gives 0 right?

so now since i have a time varying current, what should gauss law give?

thanks!
 
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  • #2
A (long) straight wire can carry a current (steady or oscillating) and still have a zero net charge distribution throughout.

A Caveat:

If you consider the end points of the wire, this may not be the case. Charge will build up at the ends, and in the case of an harmonically varying current, the charge on the end points will also vary harmonically, but out of phase with the current by [itex]\pi /2[/itex].

In this case, your charge distribution will consist of two delta functions at the ends of the wire, whose amplitudes oscillate in time.

So, if you're wire ends are at +-z:

[tex]\rho(r)= q_o \cos(\omega t) \delta(z) - q_o \cos(\omega t) \delta(-z)[/tex]
 
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  • #3
G01 said:
A wire can carry a current and still have a zero net charge distribution throughout.

issn't current the flow of charges?

how can the charged enclosed be 0?

erm, what do you mean by net charge distribution?
 
  • #4
i am getting very confused :(

if i apply a gaussian cylindrical surface to the wire, then the flux is 0?

so my net charge is 0? so i assume its the electrons and +ve charges that do the cancelling?
 
  • #5
G01 said:
A (long) straight wire can carry a current (steady or oscillating) and still have a zero net charge distribution throughout.

A Caveat:

If you consider the end points of the wire, this may not be the case. Charge will build up at the ends, and in the case of an harmonically varying current, the charge on the end points will also vary harmonically, but out of phase with the current by [itex]\pi /2[/itex].

In this case, your charge distribution will consist of two delta functions at the ends of the wire, whose amplitudes oscillate in time.

:eek:

i think the caveat part is out of my scope at the moment :(
 
  • #6
quietrain said:
issn't current the flow of charges?

how can the charged enclosed be 0?

erm, what do you mean by net charge distribution?
Consider a length of the ideal wire we are considering carrying no current.

There are the same number of positive and negative charges in it. Thus it is neutral. Agreed?

Now apply a voltage so we get a current. At anyone time, the protons and electrons are still equal in number and the wire has no net (i.e. total) charge. However, the electrons are moving, but the protons are not, resulting in a net (total/overall) current. Thus, a wire can have no net charge, but still carry current. Does this make sense?
 
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  • #7
G01 said:
Now apply a voltage so we get a current. At anyone time, the protons and electrons are still equal in number and the wire has no net (i.e. total) charge. However, the electrons are moving, but the protons are not, resulting in a net (total/overall) current. Thus, a wire can have no net charge, but still carry current. Does this make sense?

ah yes i see, but in the case of a varying magnitude current as in the original question, wouldn't this mean that the electrons moving in that unit length be not equal to the number of protons which were not moving in that unit length?

for example, say the unit length has 5 protons, since now my current magnitude is varying, i will have 5,4,3,4,5 electrons at different instances of time.
so the net charge will be different?
 
  • #8
quietrain said:
ah yes i see, but in the case of a varying magnitude current as in the original question, wouldn't this mean that the electrons moving in that unit length be not equal to the number of protons which were not moving in that unit length?

for example, say the unit length has 5 protons, since now my current magnitude is varying, i will have 5,4,3,4,5 electrons at different instances of time.
so the net charge will be different?
No. You are picturing the current varying in space. This is not what we are talking about here. In this case, the current is varying in time only!

Consider one instant in time. Call it time 0. The value of the current in the wire is [itex]I_o[/itex]. All along the wire, the current value is [itex]I_o[/itex].

Now, at a later time, say time=5s, the current value is different, [itex]I_o sin(5\omega)[/itex], but yet it is the same at every point in the wire.

Thus, in both cases, the net charge distribution is still uniform and zero, the speed of the electrons in the only thing to have changed.As an aside,

If the current was varying in space, and you had "pockets" of greater current and lesser current along the wire, then, the divergence of the current would not be zero, and by the continuity equation, we would indeed have a changing charge distribution...
 
  • #9
G01 said:
No. You are picturing the current varying in space. This is not what we are talking about here. In this case, the current is varying in time only!

Consider one instant in time. Call it time 0. The value of the current in the wire is [itex]I_o[/itex]. All along the wire, the current value is [itex]I_o[/itex].

Now, at a later time, say time=5s, the current value is different, [itex]I_o sin(5\omega)[/itex], but yet it is the same at every point in the wire.

Thus, in both cases, the net charge distribution is still uniform and zero, the speed of the electrons in the only thing to have changed.


As an aside,

If the current was varying in space, and you had "pockets" of greater current and lesser current along the wire, then, the divergence of the current would not be zero, and by the continuity equation, we would indeed have a changing charge distribution...

ah i see thank you!
 

What is Gauss Law in Electrodynamics?

Gauss Law in Electrodynamics is a fundamental law in physics that relates the distribution of electric charges to the resulting electric field. It states that the total electric flux through a closed surface is equal to the total charge enclosed by that surface divided by the permittivity of free space.

How is Gauss Law used in Electrodynamics?

Gauss Law is used to calculate the electric field at a given point due to a distribution of electric charges. It allows us to simplify complex problems by using symmetry and closed surfaces to calculate the electric field.

What is the difference between Gauss Law in Electrodynamics and Gauss Law in Magnetism?

Gauss Law in Electrodynamics deals with the electric field, while Gauss Law in Magnetism deals with the magnetic field. The key difference is that electric charges are sources of electric fields, while there are no magnetic charges (monopoles) to act as sources of magnetic fields.

Can Gauss Law in Electrodynamics be applied to non-uniform electric fields?

Yes, Gauss Law in Electrodynamics can be applied to non-uniform electric fields. In this case, the electric field is calculated by dividing the charge enclosed by the surface into small pieces, and using the principle of superposition to add up the contributions from each piece.

What are the limitations of Gauss Law in Electrodynamics?

One limitation of Gauss Law in Electrodynamics is that it only applies to static electric fields. It cannot be used to calculate the electric field in cases where the charges are moving or changing over time. Additionally, Gauss Law does not take into account the effects of relativity and quantum mechanics on the behavior of electric charges.

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