29: Potential Difference and Potential near a Charged Sheet

INTRO: Let A=(x1,y1) and B=(x2,y2) be two points near and on the same side of a charged sheet with surface charge density +σ. The electric field E⃗ due to such a charged sheet has magnitude E=σ/(2ϵ0) everywhere, and the field points away from the sheet, as shown in the diagram

PART A: 

What is the potential difference 

VAB=VA−VB between points A and B?

Express your answer in terms of some or all of E, x1, y1, x2, and y2.

  SOLUTION:

Begin by considering the line integral along the path from B to A. 

The formula for potential difference, V between the two points doesn't depend on the path, lets call it l, but rather the end points, A & B. 

VAB=−∫ABE⃗ ⋅dl⃗ ,

The line integral from B=(x2,y2) to A=(x1,y1) of C⃗ =Cxı^+Cyȷ^, a constant vector field (i.e., independent of x and y), is given by

∫ABC⃗ ⋅dℓ⃗ =∫x1x2Cxdx+∫y1y2Cydy
=Cx(x1−x2)+Cy(y1−y2).

Since the charge is consistent throughout the x direction, the x component can be disregarded

This leaves us with the solution

VAB= −E(y1−y2)

−E(y1−y2

NOTE:
the expression E(y2−y1) will not yield the correct potential if you apply it to two points on opposite sides of the sheet. For example, the expression does not indicate that two points on opposite sides of the sheet and the same distance from it are at the same potential (VAB=VA−VB=0), which is clear from the symmetry of the situation. If you take care in carrying out the integration to observe the change in the direction of the electric field as you pass from one side of the sheet to the other, you will find that the potential difference between A and B is actually given by

VAB=E(|y2|−|y1|).

PART B: 

If the potential at y=±∞ is taken to be zero, what is the value of the potential at a point VA at some positive distance y1 from the surface of the sheet?

  SOLUTION:

Substitute appropriate values for VB and y2 in the equation VA−VB=E(|y2|−|y1|).

VB = 0

y2 = ±∞

VA=E(∞) = ∞

PART C: 

Now take the potential to be zero at y=0 instead of at infinity. What is the value of VA at point A some positive distance y1 from the sheet?

  SOLUTION:

Substitute appropriate values for VB and y2 in the equation VA−VB=E(|y2|−|y1|).

VB = ±∞

y2 = 0

VA= E(0 -|y1|) 

= −E⋅y1