Electrostatics of Conductors: Properties and Formulas Class 12

Taking a scenario, which is hypothetical for now, we place a bunch of electrons on a copper sphere.

In Class 12 electrostatics, you learned that like charges repel. This sphere will face the same principle, where the excess of electrons on the sphere will repel each other. They will push and move around until they find their resting space across different spots. There will be a balance, which, according to physicists, is known as electrostatic equilibrium.
This is a basic property to take note of when you are learning the behaviour of a conductor in electrostatics. The golden rule to remember is that in a conductor at electrostatic equilibrium, the electric field is zero. If this is not zero, the free electrons feel a force (F = qE) that makes them move again. For equilibrium to happen, the field must be zero to make F = 0. 

You can tell from the conductor and insulator difference when there’s free flow of electrons in some materials like copper and aluminium, and when some, like plastic or wood, don’t allow the flow at all. In the Electrostatic Potential and Capacitance chapter, we approach the definition of a conductor in the ‘static’ state when there’s no current flowing and no charges zipping around. Everything settles into equilibrium. 

There are six important properties of conductors when in their calm state. Before going ahead with the reasoning for each, here is a quick reference table for your CBSE boards. 

Below there are explanations of all the properties, discussing the reasons why they behave the way they do. 

Know why this happens from a Physics perspective below. 

1. In equilibrium, nothing moves. There’s no force that the free electrons are going to face to move. That’s why, there seems to be no electric field inside.
2. In reality, the free-to-move charges get redistributed by themselves in such a way that their cumulative electric field can cancel out any external field that tries to enter or penetrate the conductor. 
3. When an external field tries to enter the conductor, the electrons existing near the surface run to the other side and that leaves the positive charges behind. The charges that redistribute upon contact with the electric field create their own field and that points to the external field. Inside the conductor, the two fields perfectly cancel out. 

So, electric field inside zero, ie., E_inside = 0.

The surface field is always perpendicular to the conductor in equilibrium state is an essential property to learn. It tells what happens on the surface of the conductor when the field inside is cancelled out. See the reasons why the field must be perpendicular, below. 

1. If you think the opposite, where the field is parallel to the conductor’s surface, it becomes a tangential component that moves the charges along the surface. This would lead the charges to slide past and that goes against the first property where there is no electric field inside. 
2. The only way that a perpendicular component doesn’t affect or violate the equilibrium is that it does not let the charges slide. The normal (perpendicular component) is also only possible in this case because it does not let the charges to slide 
3. Because it can only push the electrons into the conductor or pull them away from it. And in both cases, the charges would remain stationary. As is, they will be in equilibrium. 

Mathematically, E_tangent = 0

To understand this property in the electrostatics of a conductor, we will prove using previously learnt concepts. 

1. Had there been an excess charge inside the conductor, there would be an electric field and since on the inside there is no field, nothing can exist inside a conductor. They can stay on the surface, more like on the outer shell. 
2. This claim can be proven with Gauss’ Law. If you draw a Gaussian surface inside the conductor and, of course, smaller than it, the electric flux through this surface is zero as there is no electric field inside anyway.
   
   ∮ E.da = Q_enclosed/ε_0 

Now since flux has a zero value on the left side of the equation, it won’t allow for any charge to mathematically exist or have an impact inside the Gaussian surface. 

So, we can use Q_enclosed = 0.

But if at all, any excess charges exist, they must reside on the surface of the conductor in the equilibrium state. 

While you read about the electrostatic potential definition earlier, you should now be able to connect why this value of V remains constant below. 

1. Assume that inside a conductor, there are two different potentials at two different points. Like an arbitrary point A would have more potential than the one at B. And previously, you learned about equipotential surfaces, where the field directs or goes from higher to lower potentials. But since this is a conductor we’re talking about, there will be current created. And we are talking about the equilibrium state where current is not possible to exist. That means the potential must be the same throughout.
2. Even if you are considering the maths behind this, you have E = -dV/dl, where dV is the slight change in potential between two points and dl is the small reference distance to calculate the electric field. But E = 0, which means -dV/dl = 0. So dV will lead to a zero result, describing that the potential remains constant. 

What’s essential to note here is that the whole conductor, including the surface, has an equipotential volume.