Electric field within solenoid

[blintaro]

Hello all,

I can't figure out why it is impossible to have a radial electrical field within a solenoid. My gut tells me there would be one. For that matter, I also don't understand why there is no radial component surrounding a current carrying wire. Considering gauss' law has not helped me so far since I can't restrict the radial and longitudinal components of flux. I also tried to consider faraday's law by reasoning that the product (b field) of the curl must be along the axis (z direction) in cylindrical coordinates but that just created more conditions on the components of E.

If anyone can help it would be really appreciated, I might be a little butthurt since I spent the majority of my exam time puzzling over this an consequently ran out of time. Whoops!

 

[axmls]

Does a current carrying wire have a net charge? Is the magnetic field changing in time?

 

[Charles Link]

The Maxwell's equations for the electric field $E$ need to be considered. One is $\nabla \cdot E=\rho/\epsilon_o$. With Gauss's law, this becomes in integral form $\int E \cdot dA=Q/\epsilon_o$. The current carrying solenoid has radial symmetry and the enclosed $Q$ is zero. This equation tells us there will be no radial electric field $E$ inside a solenoid, even for a time-varying current. The other Maxwell equation involving $E$ is $\nabla \times E=-dB/dt$ which in integral form using Stokes theorem becomes $\oint E \cdot dl=-d(BA)/dt$ where $BA$ is the magnetic flux across the area of the path of the line integral . For the steady state case, this gives $E=0$ for the $\phi$ component, but for a time varying solenoid current, the changing $B_z$ will result in a $\phi$ component of $E$ which is a Faraday EMF inside the solenoid (as well as outside of the solenoid). Perhaps this helps to answer your question. Meanwhile there are no electrical charges to generate any static $E_z$. Even for a time-varying current, I believe $E_z=0$, but this last one is a minor detail that can be worked out by considering another of Maxwell's equations $\nabla \times B=\mu_o J +\mu_o \epsilon_o dE/dt$.

 

[blintaro]

That's a good question! Since the definition of current is time varying charge I don't see why it can't be modeled as a line of moving charge. Which can be integrated along a line at given moment of time to give some amount of net charge.

In this case the magnetic field in the solenoid was varying because the current around the solenoid was dependent on time.

 

[blintaro]

Ah I see! The enclosed charge must be zero! What about when the Gaussian surface encloses a line of current, then there might be enclosed charge right?

 

[Charles Link]

Ah I see! The enclosed charge must be zero! What about when the Gaussian surface encloses a line of current, then there might be enclosed charge right?

Normally a line of current in a conductor is still electrically neutral. The net charge contained in any portion of the conductor is considered to be zero. In some very advanced treatments of the subject, there may be some non-neutral cases considered, e.g. the Hall effect, but for most elementary cases, the current carrying conductor is assumed to be electrically neutral.

 

[blintaro]

I think that makes sense in terms of an ionic fluid or free electrons in a metal... So does that mean if a wire connected a charged sphere to an uncharged sphere, say, then there would be quantitatively different behavior?

 

[Delta2]

There will be radial component inside the solenoid , though very small, and that is due to imperfection in the symmetry (a solenoid is a helix, a not a series of perfect cyclic currents closely spaced together as we theoretically view it).

There will also be radial component in the case of a straight wire carrying current, and that is because - though the conductor will have zero net charge as a whole- of the local surface charges that are present. But these surface charge densities are small also so the radial component will be very small, practically zero.