Question about understanding conductors for EM course

[Sho Kano]

So in conductors, the electrons will distribute themselves to the surface via repulsion forces. But why do we say that the electric field inside is zero? If I put a positive charge inside, clearly it will move in some direction from the electric field of generated from the electrons. Also, are the electrons actually stationary on the surface (i.e. have a net zero force due to each other), or are they moving but have a net effect of zero?

 

[houlahound]

It appears you are conflating two different experiments/conditions?

 

[Dale]

But why do we say that the electric field inside is zero?

It is only zero in the electrostatic condition. I.e. given that a system is electrostatic then $J=0$. In a conductor $J=\sigma E$ so $E=0$.

 

[Sho Kano]

It is only zero in the electrostatic condition. I.e. given that a system is electrostatic then $J=0$. In a conductor $J=\sigma E$ so $E=0$.

Okay, so the electrons repel each other by repulsion due to each other's E fields, then end up stationary at the surface of the conductor. At this point, the parallel E fields along the surface cancel out to 0, and all that's left are the perpendicular ones. So since there are no electrons below the surface, there is no E field below the surface?

 

[Drakkith]

The electrons only arrange themselves on the surface if there are excess electrons. IE, if the conductor is negatively charged. If your conductor is an infinitely long cylindrical wire, the e-field from the electrons cancels out in every direction except radially outwards from the surface. Inside the e-field cancels out in all directions, including radially. If it didn't cancel out, you'd have an e-field set up inside the conductor and the charges would move until the e-field is zero.

 

[Sho Kano]

The electrons only arrange themselves on the surface if there are excess electrons. IE, if the conductor is negatively charged.

So in a neutrally charged conductor, the electrons are still attached to the atoms?

 

[Sho Kano]

If your conductor is an infinitely long cylindrical wire, the e-field from the electrons cancels out in every direction except radially outwards from the surface

They could cancel out like this? i.e. field lines bend upwards- won't the bottom parts contribute to a net downwards field too?

 

[Sho Kano]

Alright I understand now; There is a net zero electric field inside a conductor in static equilibrium. If you bring in an electron, it will throw off all the other electrons, and they will rearrange themselves. On a related question, shouldn't the electric field right on the surface of a conductor be zero?

Shouldn't there be a discontinuity at x=r?

 

[Drakkith]

So in a neutrally charged conductor, the electrons are still attached to the atoms?

Most are, yes. Some are free and move about the conductor. The exact number depends on the material the conductor is made out of. While these free electrons are able to move around, their net motion cancels out overall, so you don't see current flow or charged areas in a neutral conductor.

They could cancel out like this? i.e. field lines bend upwards- won't the bottom parts contribute to a net downwards field too?

That only works for two charges. There are MANY individual charges. So many that there isn't anywhere for the field lines to "bunch up" like you see in that picture.

On a related question, shouldn't the electric field right on the surface of a conductor be zero?

I don't think so. On the surface means that it isn't inside the conductor, so the electric field shouldn't be zero there.

 

[Sho Kano]

So only the excess (net charge) distributes themselves on the surface, while the net charge inside is 0. This is why there is no net E field in the conductor.

The E field inside a neutral conductor is also 0 because there is a net zero charge meaning a net zero field.

 

[Dale]

Again, only in the electrostatic case.

 

[Sho Kano]

That only works for two charges. There are MANY individual charges. So many that there isn't anywhere for the field lines to "bunch up" like you see in that picture.

Yes. Looking at this picture though, there seems to be also perpendicular field lines extending into the conductor (from the surface charges)- do these cancel out each other too?

 

[Sho Kano]

Again, only in the electrostatic case.

Great!

 

[Sho Kano]

I don't think so. On the surface means that it isn't inside the conductor, so the electric field shouldn't be zero there.

If I try to apply Gauss's law here, the Gaussian surface will be right on the charges, so there seems to be no enclosed charge?

 

[pixel]

If I try to apply Gauss's law here, the Gaussian surface will be right on the charges, so there seems to be no enclosed charge?

The "surface" is an idealization - something that is infinitely thin and yet contains any excess charges assumed to be present. What happens exactly at the surface is more of a philosophical question. For instance, if the surface is "right on the charges," then some parts of the charges are within the surface and some parts are outside. That would lead to a transition regime to replace the vertical dashed line in your plot.

 

[Sho Kano]

The "surface" is an idealization - something that is infinitely thin and yet contains any excess charges assumed to be present. What happens exactly at the surface is more of a philosophical question. For instance, if the surface is "right on the charges," then some parts of the charges are within the surface and some parts are outside. That would lead to a transition regime to replace the vertical dashed line in your plot.

How do we calculate the field at a point on the surface? So would Gauss's law still hold, and we would use the radius of the sphere?

 

[Dale]

It is discontinuous on the surface.

 

[pixel]

It is discontinuous on the surface.

Yes, over any reasonable distance scale, but the OP is asking - maybe unnecessarily - about the E field in a small region at the "surface."

 

[Sho Kano]

Right, it's much clearer now. A quick question, why do we experience electric shocks (i.e. when touching metals) if the human skin is not a great conductor? Let's say I build up a net positive charge from friction, then approach a doorknob. My finger's electric field will induce a negative charge on the doorknob, and when I touch it the electrons flow through my finger, and I feel a shock. Skin is not a great conductor, also the rubber shoes are not great either.

 

[Sho Kano]

That only works for two charges. There are MANY individual charges. So many that there isn't anywhere for the field lines to "bunch up" like you see in that picture.

Okay I understand now, but one thing is still incomplete- the field lines would also point inwards as well as outwards. These inward lines add up to a field inside right?

 

[Drakkith]

Right, it's much clearer now. A quick question, why do we experience electric shocks (i.e. when touching metals) if the human skin is not a great conductor? Let's say I build up a net positive charge from friction, then approach a doorknob. My finger's electric field will induce a negative charge on the doorknob, and when I touch it the electrons flow through my finger, and I feel a shock. Skin is not a great conductor, also the rubber shoes are not great either.

Skin and shoes may not be great conductors, but when the voltage between your finger and the doorknob is high enough to ionize the air itself (resistance of around 10¹³ ohms/mm) the 100,000 ohm resistance of your skin does very little. Also, charge isn't be transferred through your shoes when you shock yourself on a doorknob, you yourself have already accumulated a charge.

Okay I understand now, but one thing is still incomplete- the field lines would also point inwards as well as outwards. These inward lines add up to a field inside right?

Imagine you keep adding charges to your picture above. The field lines coming from each electron would be shoved closer and closer together until they just radiate straight outwards, away from the center of the wire. The field inside the wire is zero, meaning that a charge of either type placed in the middle of the wire feels no net force in any direction. This is just like a slice through a hollow, charged sphere.

 

[Sho Kano]

Also, charge isn't be transferred through your shoes when you shock yourself on a doorknob, you yourself have already accumulated a charge.

What are the situations where charge goes into you and directly to the ground? High enough voltage?

Imagine you keep adding charges to your picture above. The field lines coming from each electron would be shoved closer and closer together until they just radiate straight outwards, away from the center of the wire.

It is still a bit hard to imagine. Yes the field lines would radiate straight outwards, but it seems like the same would happen inwards even if I bunch the charges up (i.e. field lines radiate straight inwards). But since we know the electric to be zero inside a conductor, what happens to these inward lines?

 

[Drakkith]

What are the situations where charge goes into you and directly to the ground? High enough voltage?

No, I could apply 120 volts between your foot and your hand and get current flow. My point was that when you shock yourself your shoes don't afford any protection because your body has already developed a charge.

It is still a bit hard to imagine. Yes the field lines would radiate straight outwards, but it seems like the same would happen inwards (i.e. field lines radiate straight inwards). But since we know the electric to be zero inside a conductor, what happens to these inward lines?

There aren't any. They would all radiate outwards. Remember that field lines are visualization tools that help us visualize the magnitude and direction of the force exerted on a charged particle if placed in that field. They are not a fundamental explanation in and of themselves. If you place a charged particle inside your conductor and then calculate the net force from every single electron you would find that the net force is zero no matter where you place the charge inside the conductor.

 

[Sho Kano]

Remember that field lines are visualization tools that help us visualize the magnitude and direction of the force exerted on a charged particle if placed in that field. They are not a fundamental explanation in and of themselves.

But field lines are accurate representations of the electric field, and the electric field gives an accurate representation of the forces on an charged particle.

There aren't any. They would all radiate outwards.

I can't visualize why there wouldn't be an electric field inwards. Maybe I have to just downright accept it?

 

[Drakkith]

But field lines are accurate representations of the electric field, and the electric field gives an accurate representation of the forces on an charged particle.

Of course. And if the electric field inside the conductor is zero, then there aren't any field lines.

I can't visualize why there wouldn't be an electric field inwards. Maybe I have to just downright accept it?

Let me ask you this. In the picture above, what is in between any two field lines? Do charged particles have to be placed on a field line in order to feel a force?

 

[Sho Kano]

Do charged particles have to be placed on a field line in order to feel a force?

The picture is just a simplification, there are field lines everywhere in space. But if we bunch up a whole lot of them in a line, there will be a net upwards field- And also a net downwards field. The situation doesn't seem to change if we bunch them up along a circle. So in the situation of a circle, we are left with field lines pointing towards the center of the circle.

 

[pixel]

Right, it's much clearer now. A quick question, why do we experience electric shocks (i.e. when touching metals) if the human skin is not a great conductor? Let's say I build up a net positive charge from friction, then approach a doorknob. My finger's electric field will induce a negative charge on the doorknob, and when I touch it the electrons flow through my finger, and I feel a shock. Skin is not a great conductor, also the rubber shoes are not great either.

Charge builds up on your skin. Before touching the metal, it jumps across the air making a spark. It's less prevalent in humid weather as it's not as easy for the air to break down.

 

[davenn]

It's less prevalent in humid weather as it's not as easy for the air to break down.

That is not the reason
The reason is that on humid days things don't hold a charge so easily as the humid air allows it to leak away
rather that build upDave

 

[Drakkith]

The picture is just a simplification, there are field lines everywhere in space. But if we bunch up a whole lot of them in a line, there will be a net upwards field- And also a net downwards field. The situation doesn't seem to change if we bunch them up along a circle. So in the situation of a circle, we are left with field lines pointing towards the center of the circle.

Well, you can see for yourself from your picture that the lines do indeed bunch up and curve away from the lines of other like charges. We can add as many lines to each charge as we like and this will still be true. If we start adding more charges into our picture, putting them in the shape of a circle, you'll find that the field lines are bent even more strongly. Add enough charges, so many that we can stop thinking about them as discrete objects and more like a continuous circle of charge, and you'll find that the field lines no longer go inward at all. And how could they? If we have so many charges that we can treat them as continuous, any field line inside the conductor would start and end on the same type of charge, which isn't possible.

 

[pixel]

That is not the reason
The reason is that on humid days things don't hold a charge so easily as the humid air allows it to leak away
rather that build up.

That's true, but it's also true that for a given amount of charge on your hand the ability to ionize the air depends on the humidity level. Water is a polar molecule and the water molecules line up in a manner to reduce the potential difference between your hand and the metal.