# GAUSS’S LAW

## Conductors in Electrostatic Equilibrium

We know that a good electrical conductor contains charges (electrons) that are not bound to any atom and therefore are free to move about within the material. When there is no net motion of charge within a conductor, the conductor is in electrostatic equilibrium. As we shall see, a conductor in electrostatic equilibrium has the following properties:

- The electric field is zero everywhere inside the conductor.
- If an isolated conductor carries a charge, the charge resides on its surface.
- The electric field just outside a charged conductor is perpendicular to the surface of the conductor and has a magnitude σ/ε
_{0} , where σ is the surface charge density at that point.
- On an irregularly shaped conductor, the surface charge density is greatest at locations where the radius of curvature of the surface is smallest.

We verify the first three properties in the discussion that follows. The fourth property is presented here without further discussion so that we have a complete list of properties for conductors in electrostatic equilibrium.

We can understand the first property by considering a conducting slab placed in an external field E (Fig. 24.16). We can argue that the electric field inside the conductor must be zero under the assumption that we have electrostatic equilibrium. If the field were not zero, free charges in the conductor would accelerate under the action of the field. This motion of electrons, however, would mean that the conductor is not in electrostatic equilibrium. Thus, the existence of electrostatic equilibrium is consistent only with a zero field in the conductor.

Let us investigate how this zero field is accomplished. Before the external field is applied, free electrons are uniformly distributed throughout the conductor. When the external field is applied, the free electrons accelerate to the left in Figure 24.16, causing a plane of negative charge to be present on the left surface. The movement of electrons to the left results in a plane of positive charge on the right surface. These planes of charge create an additional electric field inside the conductor that opposes the external field. As the electrons move, the surface charge density increases until the magnitude of the internal field equals that of the external field, and the net result is a net field of zero inside the conductor. The time it takes a good conductor to reach equilibrium is of the order of 10^{–16} s, which for most purposes can be considered instantaneous.

We can use Gauss’s law to verify the second property of a conductor in electrostatic equilibrium. Figure 24.17 shows an arbitrarily shaped conductor. A Gaussian surface is drawn inside the conductor and can be as close to the conductor’s surface as we wish. As we have just shown, the electric field everywhere inside the conductor is zero when it is in electrostatic equilibrium. Therefore, the electric field must be zero at every point on the gaussian surface, in accordance with condition (4) in previous topic. Thus, the net flux through this gaussian surface is zero. From this result and Gauss’s law, we conclude that the net charge inside the gaussian surface is zero. Because there can be no net charge inside the gaussian surface (which is arbitrarily close to the conductor’s surface), any net charge on the conductor must reside on its surface. Gauss’s law does not indicate how this excess charge is distributed on the conductor’s surface.

We can also use Gauss’s law to verify the third property. We draw a Gaussian surface in the shape of a small cylinder whose end faces are parallel to the surface of the conductor (Fig. 24.18).

Part of the cylinder is just outside the conductor, and part is inside. The field is normal to the conductor’s surface from the condition of electrostatic equilibrium. (If E had a component parallel to the conductor’s surface, the free charges would move along the surface; in such a case, the conductor would not be in equilibrium.) Thus, we satisfy condition (3) in last topic for the curved part of the cylindrical gaussian surface - there is no flux through this part of the gaussian surface because E is parallel to the surface. There is no flux through the flat face of the cylinder inside the conductor because here E = 0 - satisfaction of condition (4). Hence, the net flux through the Gaussian surface is that through only the flat face outside the conductor, where the field is perpendicular to the gaussian surface. Using conditions (1) and (2) for this face, the flux is EA, where E is the electric field just outside the conductor and A is the area of the cylinder’s face. Applying Gauss’s law to this surface, we obtain

Where we have used the fact that q_{in} = σA. Solving for E gives

Next Topic:- Experimental Verification of Gauss’s Law and Coulomb’s Law