ELECTROSTATICS OF CONDUCTORS
In a metal, the outer (valence) electrons part away from their atoms and are free to move. These electrons are free within the metal but not free to leave the metal. The free electrons form a kind of gas, they collide with each other and with the ions, and move randomly in different directions. The positive ions made up of the nuclei and the bound electrons remain held in their fixed positions.
Let us note important results regarding
electrostatics of conductors.
`text(1. Inside a conductor, electrostatic field is zero)`
Consider a conductor, neutral or charged. There may also be an external electrostatic field. In the static situation, when there is no current inside or on the surface of the conductor, the electric field is zero everywhere inside the conductor. This fact can be taken as the defining property of a conductor. A conductor has free electrons. As long as electric field is not zero, the free charge carriers would experience force and drift. In the static situation, the free charges have so distributed themselves that the electric field is zero everywhere inside. Electrostatic field is zero inside a conductor.
`text(2. At the surface of a charged conductor, electrostatic field must be normal to the surface at every point)`
If E were not normal to the surface, it would have some non-zero component along the surface. Free charges on the surface of the conductor would then experience force and move. In the static situation, therefore, E should have no tangential component. Thus electrostatic field at the surface of a charged conductor must be normal to the surface at every point. (For a conductor without any surface charge density, field is zero even at the surface.).
`text(3. The interior of a conductor can have no excess charge in the static situation)`
A neutral conductor has equal amounts of positive and negative charges in every small volume or surface element. When the conductor is charged, the excess charge can reside only on the surface in the static situation. This follows from the Gauss-s law. Consider any arbitrary volume element v inside a conductor. On the closed surface S bounding the volume element v, electrostatic field is zero. Thus the total electric flux through S is zero. Hence, by Gauss-s law, there is no net charge enclosed by S. But the surface S can be made as small as you like, i.e., the volume v can be made vanishingly small. This means there is no net charge at any point inside the conductor, and any excess charge must reside at the surface.
`text(4. Electrostatic potential is constant throughout the volume of the conductor and has the same value on its surface)`
This follows from results 1 and 2 above. Since E = 0 inside the conductor and has no tangential component on the surface, no work is done in moving a small test charge within the conductor and on its surface. That is, there is no potential difference between any two points inside or on the surface of the conductor.
`text(5.Electric field at the surface of a charged conductor)`
where σ is the surface charge density and -n is a unit vector normal to the surface in the outward direction.
To derive the result, choose a pill box (a short cylinder) as the Gaussian surface about any point P on the surface, as shown in Fig. The pill box is partly inside and partly outside the surface of the conductor. It has a small area of cross section `δS` and negligible height.
`E = |sigma|/epsilon_0`
Including the fact that electric field is normal to the surface, we get the vector relation, which is true for both signs of σ. For σ > 0, electric field is normal to the surface outward; for σ < 0, electric field is normal to the surface inward.
Let us note important results regarding
electrostatics of conductors.
`text(1. Inside a conductor, electrostatic field is zero)`
Consider a conductor, neutral or charged. There may also be an external electrostatic field. In the static situation, when there is no current inside or on the surface of the conductor, the electric field is zero everywhere inside the conductor. This fact can be taken as the defining property of a conductor. A conductor has free electrons. As long as electric field is not zero, the free charge carriers would experience force and drift. In the static situation, the free charges have so distributed themselves that the electric field is zero everywhere inside. Electrostatic field is zero inside a conductor.
`text(2. At the surface of a charged conductor, electrostatic field must be normal to the surface at every point)`
If E were not normal to the surface, it would have some non-zero component along the surface. Free charges on the surface of the conductor would then experience force and move. In the static situation, therefore, E should have no tangential component. Thus electrostatic field at the surface of a charged conductor must be normal to the surface at every point. (For a conductor without any surface charge density, field is zero even at the surface.).
`text(3. The interior of a conductor can have no excess charge in the static situation)`
A neutral conductor has equal amounts of positive and negative charges in every small volume or surface element. When the conductor is charged, the excess charge can reside only on the surface in the static situation. This follows from the Gauss-s law. Consider any arbitrary volume element v inside a conductor. On the closed surface S bounding the volume element v, electrostatic field is zero. Thus the total electric flux through S is zero. Hence, by Gauss-s law, there is no net charge enclosed by S. But the surface S can be made as small as you like, i.e., the volume v can be made vanishingly small. This means there is no net charge at any point inside the conductor, and any excess charge must reside at the surface.
`text(4. Electrostatic potential is constant throughout the volume of the conductor and has the same value on its surface)`
This follows from results 1 and 2 above. Since E = 0 inside the conductor and has no tangential component on the surface, no work is done in moving a small test charge within the conductor and on its surface. That is, there is no potential difference between any two points inside or on the surface of the conductor.
`text(5.Electric field at the surface of a charged conductor)`
where σ is the surface charge density and -n is a unit vector normal to the surface in the outward direction.
To derive the result, choose a pill box (a short cylinder) as the Gaussian surface about any point P on the surface, as shown in Fig. The pill box is partly inside and partly outside the surface of the conductor. It has a small area of cross section `δS` and negligible height.
`E = |sigma|/epsilon_0`
Including the fact that electric field is normal to the surface, we get the vector relation, which is true for both signs of σ. For σ > 0, electric field is normal to the surface outward; for σ < 0, electric field is normal to the surface inward.