Solution to Image Charge Method for Two Point Charges

In summary, two point charges of equal value are placed a distance 2b from each other in the presence of a grounded conducting sphere. The minimum radius of the sphere required to cancel out the repelling force between the two charges can be solved using the Image Charge Method. By adding a negative charge at a distance R^2/r from the center of the sphere in the positive x-direction, and another negative charge at a distance R^2/r from the center of the sphere in the negative x-direction, the potentials from the two configurations can be added up. The equation for R/d can only be solved numerically and there is only one value of R that causes equilibrium.
  • #1
Yosty22
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Homework Statement


Two point charges of equal value are placed a distance 2b from one another. In the middle, there is a grounded conducting sphere. What is the minimum radius of the sphere required to cancel out the repelling force between the two charges? (Solve using Image Charge Method)

Homework Equations


The Attempt at a Solution



I attached a file I used to create the scenario in. During one of the lectures, we proved what I wrote on the right-hand side of the file to show that in order to keep the conducting sphere at a potential of 0, you must put a negative charge at a distance R^2/r from the center of the sphere. Since this is spherically symmetric and you are adding another positive charge outside of the sphere (the charge labeled q2), could you just add another mirrored charge a distance R^2/r from the center of the sphere, except this time, in the negative direction? That is, add a charge of -q at a position R^2/r to the left of the center.

Therefore, your total setup for the mirrored charges would be one charge of q = -q located a distance R^2/r from the center of the sphere in the positive x-direction and another charge q = -q' located a distance R^2/r from the center of the sphere in the negative x-direction?
 

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  • #2
I agree with the approach: the potentials from the two configurations can be added up.
I miss the relevant equation, but never mind.
Symmetry isn't spherical but rotational. Plus there is mirror symmetry. Never mind that either.
Given is that the charges are equal, makes is somewhat more manageable.
So q1 = q2 = q.
They are at distance 2b which became 2d. Fine too.

Mirror charge value is also known; it's not -q.

All in all you have four charges lined up at -d, -R2/d, R2/d, and d.
Your job is to say something about R in the case the field at -d and +d has to be zero.

That's where I end up with a hefty equation for R/d that I can only solve numerically. And why the exercise says minimum R is weird to me: there is only one R that causes this equilibrium.

Please check and correct me :)
 

FAQ: Solution to Image Charge Method for Two Point Charges

What is the Image Charge Method for Two Point Charges?

The Image Charge Method is a mathematical technique used to analyze the electric potential and electric field created by two point charges in the presence of a conductive boundary. It involves creating a "mirror" charge on the opposite side of the boundary to account for the influence of the original charges.

How does the Image Charge Method work?

The Image Charge Method works by applying the principle of superposition, which states that the total electric potential and electric field at a point is the sum of contributions from all individual charges. By using a mirror charge, the influence of the original charges on the conductive boundary is cancelled out, allowing for easier calculations.

What are the advantages of using the Image Charge Method?

One of the main advantages of the Image Charge Method is that it simplifies the calculations required to determine the electric potential and electric field in the presence of a conductive boundary. It also allows for the visualization of the electric field lines, making it easier to understand the behavior of the charges.

Are there any limitations to the Image Charge Method?

Yes, there are some limitations to the Image Charge Method. It can only be applied in situations where the conductive boundary is a perfect conductor and the charges are point charges. It also assumes that the boundary is infinitely large, which may not always be the case in real-world scenarios.

How is the Image Charge Method used in practical applications?

The Image Charge Method is commonly used in physics and engineering to solve problems involving conductive boundaries and point charges. It has applications in fields such as electrostatics, electromagnetism, and electrochemistry. It is also used in the design of electronic devices and circuits that involve conductive boundaries.

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