Physics Course Help With Potential inside the Conductor
10.15 Potential inside the Conductor
We know that E = -dV/dr. If everywhere inside the conductor, then the potential V should either be zero, or should have some constant value for all points inside the conductor. Thus, a conductor in an electrostatic field provides an equipotential region (whole of its inside). Even its surface is an equipotential surface.
Excess Charges Inside the Conductor
Since everywhere inside the conductor, the Gauss's theorem demands that the density of excess (uncompensated) charge inside the conductor is equal to zero at all points (r = 0). There are no excess charges inside the conductor.
Field Lines at the Surface of the Conductor
The fact that the surface of a conductor is equipotential, implies that the field in the immediate vicinity of the surface must be directed along the normal to the surface. That is, the field lines are perpendicular to a conducting surface. If the opposite were true, the tangential component of would make the charges move over the surface of the conductor, and the charge equilibrium would never be reached.
Excess Charges at the Surface of the Conductor
The excess charges appear only on the surface of the conductor placed in an electric field. There is certain surface charge density s, which is generally different for different points of the surface.
In fact, the excess charge is located in a very thin surface layer (only one or two atomic layers).
Modification of Field due to Present of a Conductor
The figure shows the modified electric field, when a conductor (say, a sphere) is placed in a uniform field. Charges are induced in the conductor (which resides on its surface). These induced charges produce an additional field, which interacts with the original field. The field is thus modified. The field lines are normal to the surface of the conductor. The equipotential lines (dashed) are perpendicular to the field lines everywhere.
An isolated charged sphere creates an electric field, which is spherically symmetric. All the equipotential surfaces are concentric spheres. The field lines uniformly emerge away from the sphere in all directions.
Field Intensity near a Conductor Surface
The electric field intensity, E in the immediate vicinity of the surface of a conductor is given as
where s is the local surface charge density at that point.
If, at any point, s > 0, then is directed away from the conductor surface. And if s < 0, is directed towards the conductor surface.
Force Acting on the Surface of a Conductor
When a conductor is placed into an electric field, or when a certain charge is imported to it, charges appear on the surface of the conductor. The surface charge density s may or may not be same everywhere on the surface.
Consider a small section of the conductor near its surface. The force on small area element DS on the surface is
where s DS is the charge of this element and is the field created by all other charges of the system.
Note that is not equal to the net field intensity existing in the vicinity of the surface element.
Let = the intensity of the field created by the charge of area element DS.
= the intensity of the field created by all the other charges of the system
and = the intensity of the resultant field when the fields and are superimposed.
At the points very close to area element, it behaves as an element, it behaves as an infinite uniformly charged plane. Hence,
on the two sides of the area element, Es has opposite directions; but E0 has same direction.
Inside the conductor, <, Es = E0. Therefore, outside the conductor,
E = E0 + Es = 2E0.
Thus, Eqn. (i) becomes
Dividing both sides by DS, we obtain the force acting on unit surface, called surface density of force,
Since , at any point just outside the surface, we have
= (as s = e0 E)
This shows that regardless of the sign of s (and hence the direction of , the force is directed outside the conductor. That is, the force tends to stretch the conductor.
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