How Can the Potential Due to an Infinitely Large Plate Be Justified?

[pardesi]

i was going through the proof of the classical image problem in which u have agrounded conductor and you have a charge q above it and u r asked to find potenntial at all points above the conductor the proof uses the fact that the potential due to the sheet at "all" infinities is 0.how does one justify this

 

[olgranpappy]

If the potential goes to zero at infinity as approached in one direction then it goes to zero in all directions because outside the region containing the charge (far from the plate and the pt charge) I am just solving the Laplace equation. Thus, the angular dependence is given by Legendre polynomials and the radial dependence is at most 1/r. Since the Legendre polynomials never blow up the potential falls off at least as fast as 1/r regardless of the direction.

 

[pardesi]

what about a point infinitely separated from the point charge but just above the plate

 

[walkinginwater]

what about a point infinitely separated from the point charge but just above the plate

it is also zero!

 

[pardesi]

it is also zero!

well how is that proved?

 

[walkinginwater]

well how is that proved?

Because of the symmetry, the field line above the plate will be upwards. Assume you move the electron along the field lines. The electron will always accelerated along the field lines, i.e., the potential will decrease. The key issue here is the word "infinitely", it means that you can move an electron along the field lines infinitely until the potential decrease to zero!

 

[pardesi]

yes far above the plates that is true but what i asked before and now is a point far from the point cahrge but just close to the plate

 

[pardesi]

and one more thing adding to the problem is the situation is highly unsymmetrical

 

[walkinginwater]

and one more thing adding to the problem is the situation is highly unsymmetrical

hi, Pardesi:
A point far from the point charge means that the electric potential caused by the point charge can be neglected; close to the plate means that its potential is near the potential of the plate. so it is also zero
The key point is that the potential caused by the point charge is inversely proportional to the distance from the charge. So basically , at the infinitely far away from the particle charge, the potential due to the particle charge is zero.

 

[CompuChip]

It doesn't matter. The potential of a point charge is $$\propto 1/r$$; so it drops to zero at infinity and therefore so does the (potential of the) induced charge in the plate.

 

[pardesi]

so if i am not wrong the fundamental point is inverse propotionality of v with r and the continuity of v

 

[walkinginwater]

so if i am not wrong the fundamental point is inverse propotionality of v with r and the continuity of v

Ja, you got the point!

 

[pardesi]

thank u all