Why No Net Electric Field Inside a Conducting Sphere?

[UMath1]

I don't understand why there is no net electric field inside a conductor. Let's say the conductor is a sphere with excess charge. The charges will then distribute themselves towards the edges. Now, if I place a test charge anywhere except for the center of the sphere, wouldn't there be a net field? It would be closer to some charges and farther away from others.

 

[mfb]

Now, if I place a test charge anywhere except for the center of the sphere, wouldn't there be a net field?

No.
It is closer to a smaller amount of charge and farther away from a larger amount of charge. You can calculate that with an integral, or use Gauß' law and the spherical symmetry, the field is zero.

 

[UMath1]

How would you do it with an integral?

 

[mfb]

Integrate $\frac{\vec{r}}{|r|^3} dq$ over the whole sphere. It's a standard textbook exercise, but Gauß' law is much easier.

 

[UMath1]

If there is no field inside the conductor, what provides the force for the excess charge to distribute themselves across the edges? Let's you have an arbitrary number of excess negative charges in the conductor, in this case maybe 4. If you add another negative charge, then there would be no field acting on it. But wouldn't it still move to edge? Isn't that what excess charges do?

 

[Nugatory]

If there is no field inside the conductor, what provides the force for the excess charge to distribute themselves across the edges? Let's you have an arbitrary number of excess negative charges in the conductor, in this case maybe 4. If you add another negative charge, then there would be no field acting on it.

If anything happens that starts to create a field inside the conductor (such as introducing another negative charge) then all the other charged particles, not just the excess charges, will start moving in response to that field. They can move freely because we're dealing with a conductor; naturally they follow the fieldlines from regions of high potential to regions of low potential, thereby reducing the potential difference between regions. Thus, as long as there is any potential difference anywhere in the conductor, the charges will move in a way that reduces that difference. They don't stop moving until the entire interior of the conductor is back to being an equipotential - no potential differences, no electric field.

The short way of saying the same thing: There cannot be any electrical field in a conductor because if there all the charged particles would rearrange themselves to make it go away.

 

[mfb]

If there are charges, there is a field until no charges are left, neglecting thermal fluctuations.

 

[UMath1]

No but even when there is excess charge there is supposed to be no field after charge distributes itself to the edge. But then if I place a charge in the conductor, how can it distribute itself to the edge when there is no net field inside?

 

[nasu]

If you add a charge inside the conductor (previously at equilibrium) you will create a disturbance. The filed won't be zero anymore.
If the extra charge is fixed inside, you will have a net field inside, at the new equilibrium. If the extra charge is free to move, it will move (and so will do the existing charge) until a new situation with zero field is reached.

To have zero field inside conductors it is essential to have "free" charges, like the free electrons.
If some charge is fixed, then you don't have the conditions for zero field.

 

[ehild]

No but even when there is excess charge there is supposed to be no field after charge distributes itself to the edge. But then if I place a charge in the conductor, how can it distribute itself to the edge when there is no net field inside?

If you place a charge into the conductor it will have its field around itself. (you can imagine an ion implanted into the metal). That field interacts with the free electrons in the metal, attracts them or repulses them. At the end, the excess charge inside the conductor becomes neutralized, and an equal charge appears at the surface of the conductor.

 

[UMath1]

Can you explain the difference between the situation at 9:09 and 14:21 in this video?