Method of images& half a sphere

In summary, the problem involves finding the potential in all space for a configuration consisting of a half-sphere with charge q on its surface and a z=0 plane with charge -q, along with a charge -q' and a charge q' placed symmetrically on either side of the plane in order to cancel out the potential on the sphere. This simplifies the problem to finding the potential for a configuration of a sphere with charges q' and -q' placed symmetrically on either side of a z=0 plane.
  • #1
Ellyl
3
0

Homework Statement


[PLAIN]http://img532.imageshack.us/img532/3751/39842434.png
Find the potential in all space for this configuration.

Homework Equations


..basic electrostatics stuff, I guess.


The Attempt at a Solution


I can easily cancel the whole z=0 plane by placing a charge of -q under it, of course, but I have no idea what I should do about the half-sphere. I naturally went for placing a charge in the center of the sphere to cancel the other two, and I attempted this by calculating the potential on the top of the sphere ([PLAIN]http://img15.imageshack.us/img15/6324/kqr.gif, if pythagoras was right) and negating that with another charge, but this naturally messes the z=0 plane. Is it some other configuration of charges that's symmetrical for the z=0 plane? I guess it has to be, but I have no idea what such configuration would also negate the potential of the sphere.
 
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  • #2
..no, wait, I think I actually got this. Once I place the -q charge under the plane, the problem becomes equivalent to this:
[PLAIN]http://img189.imageshack.us/img189/6430/spherel.png
, and then I just need to place charges q' and -q' symmetrically to the z=0 plane so that the potential on the sphere equals zero. Right?
 
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1. What is the method of images?

The method of images is a mathematical technique used to solve problems involving electric charges or magnetic poles in the presence of conductive boundaries. It is based on the principle of superposition, where an imaginary charge or pole is placed at a specific location to mimic the behavior of the actual charge or pole.

2. How does the method of images work?

The method of images works by using the principle of superposition to find the electric or magnetic field at a specific point. The method involves placing an imaginary charge or pole at a specific location and calculating the resulting field. The final field is a combination of the actual and imaginary fields.

3. What is the half-sphere method of images?

The half-sphere method of images is a specific application of the method of images, where the conductive boundary is a half-sphere. This method is commonly used to solve problems involving a point charge or pole located above a grounded half-sphere.

4. What are the advantages of using the method of images?

The method of images is advantageous because it simplifies complex problems by reducing them to simpler, known solutions. It also allows for quick and efficient calculations without the need for advanced mathematical techniques. Additionally, it can be used to solve problems involving conductive boundaries, which are often difficult to solve using other methods.

5. Are there any limitations to the method of images?

Yes, there are limitations to the method of images. It can only be used for problems with simple geometries and conductive boundaries. Additionally, the method may not accurately represent the behavior of the actual system, as it relies on the approximation of an imaginary charge or pole. It also cannot be used to solve problems involving time-varying fields.

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