(Image Theory) Boundary conditions and how many image charges do i need?

[krajin]

Homework Statement

An electrostatic point charge of 1 Coulomb (C) placed symmetrically
between two infinitely/perfectly conducting parallel plates. These two infinitely large
conducting plates are parallel to the yz plane.The region between the two plates is designated as “Region A.” Starting with this original problem,we will remove the conducting plates, we will use the image theory, and we will define image charges to obtain an equivalent problem. Conducting plates coincide with planes P1 and P2, and hence these conducting plates impose some boundary conditions on the electric field.
Q1) What are the boundary conditions for the electric field on the infinitely/perfectly conducting plates P1 and P2?
We have to define a measure of approximation for Problem2(image charges). In other words, as an equivalent problem, how close is Problem 2 to Problem 1(Parallel plates)? If we use a finite number of image sources,we know that Problem 2will not be exactly equivalent to Problem 1. Nevertheless, we need to
understand how much error we are making by using a finite number of image sources.For this
purpose, let us define a measure of error as the ratio of the magnitude of the tangential
component of the electric field to the normal component of the electric field at the point E:

magnitude of the tangential component of the electric field (At the point on P1 plate)
e= -----------------------------------------------------------
electric field to the normal component of the electric field (At the point on P1 plate)

Q2) Let us consider a case where we would like to have a
maximum of 1 error. In other words, we would like to have e<0.01.
.How many image sources do we need to satisfy this constraint?

Homework Equations

E_2t = E_1t
D_1n - D_2n = ρ
E= -∇Φ
Φ = q/r

The Attempt at a Solution

For Q1 i know that E1t = E2t and as back side of the plate Et1=0=Et2 and n(normal) . D1 = ρ

For Q2 i know for not to make any error i need infinite number of image sources.

the equation Φ = qƩ(n = 0 to ∞) { (-1)ⁿ⁺¹ / [(x-nd)² + y² + z²]^1/2 + (-1)ⁿ⁺¹ / [ (x + nd)² + y² + z²]^1/2}
The attempts that i made maybe totally incorrect.

 

[anathema]

Your attempt for Q1 is correct. The boundary conditions for the electric field on the conducting plates are that the tangential component of the electric field is continuous across the boundary and the normal component of the electric field is discontinuous by an amount equal to the surface charge density (ρ).

For Q2, your understanding is partially correct. While it is true that an infinite number of image sources would result in no error, it is not practical or necessary to use an infinite number of sources.

The equation you have written for Φ is the potential due to a point charge and its image charges. However, in this problem, we are interested in the electric field, not the potential. The equation for the electric field due to a point charge and its image charges is given by:

E = qƩ(n = 0 to ∞) { (-1)n+1 / [(x-nd)2 + y2 + z2]3/2 + (-1)n+1 / [ (x + nd)2 + y2 + z2]3/2}

To find the number of image sources needed to satisfy the constraint of e<0.01, we need to use the equation for e that was given in the problem statement. Setting e<0.01, we get:

|Et|/|En| < 0.01

Using the equation for the electric field due to a point charge and its image charges, we can calculate the tangential and normal components of the electric field at a point on the conducting plate. We can then use these values to find the number of image sources needed to satisfy the constraint.

Alternatively, we can use the equation for e to find the maximum distance (dmax) between the point charge and its image sources that will result in an error of e<0.01. This can then be used to determine the number of image sources needed by dividing the total distance between the two conducting plates (which is given as infinite in the problem statement) by dmax.

I hope this helps. Let me know if you need any further clarification.