E&M Method of Images: Amplitude Adjustment?

[WastedMeat]

The method of images exploits the fact any linear combination of solutions to a differential equation is also a solution, and the fields and potentials from a point source satisfy all relevant diff eq's, assuming the boundary conditions are met.

However, a scalar multiple of any solution is also a solution, and for problems involving structures such as a semi-infinite conducting plane where the boundary conditions are that components of the electric field are zero, how is one guaranteed that the physical field amplitude is preserved after introduction of the image charges?

 

[siddharth]

However, a scalar multiple of any solution is also a solution, and for problems involving structures such as a semi-infinite conducting plane where the boundary conditions are that components of the electric field are zero, how is one guaranteed that the physical field amplitude is preserved after introduction of the image charges?

Maybe it's because one isn't arbitrarily going to use scalar multiples as we introduce image charges? By coloumb's law, there's only one way to construct the image charge, and so it doesn't make sense to introduce arbitrary multiples.

 

[WastedMeat]

1. I am not sure that there is generally only one possible image configuration for a given system, or how an argument against this can follow directly from Coulomb's law.

2. I am not saying that I should just arbitrarily multiply the fields because I have a weird fetish for such things. When you construct image charges, you are constructing a solution to Maxwell's equations that obeys the boundary conditions that you explicitly enforce. If you are not imposing any constraints on the field's amplitude, as you do not in the common example of a point charge and infinite conducting plane (aside from setting chosen amplitudes to zero), I can see no reason why the physical amplitude of the entire field configuration must be consistent with that of the image representation.

 

[siddharth]

If you are not imposing any constraints on the field's amplitude, as you do not in the common example of a point charge and infinite conducting plane (aside from setting chosen amplitudes to zero), I can see no reason why the physical amplitude of the entire field configuration must be consistent with that of the image representation.

I understand what you're saying. My point is that, if you consider the example of the point charge q and the infinite conducting plane, what you're going to do is introduce a charge of -q the same distance below the plane, so that the potential on the plane is zero.

Since it satisfies laplace's equation and the boundary condition, the solution to this problem is the unique solution to the infinite plane and the point charge. There's no freedom to introduce a scalar multiple on the potential in this case.

Similarly, for set of charges, you're always going to add the sum of the potentials. Again, the only freedom you have is in changing the strength of the charge, and if this satisfies the boundary conditions, the amplitude is preserved.

Did that help?

 

[FrogPad]

1. I am not sure that there is generally only one possible image configuration for a given system, or how an argument against this can follow directly from Coulomb's law.

My understanding is that the Uniqueness Theorem guarantees that one solution is the only solution.

http://farside.ph.utexas.edu/teaching/em/lectures/node62.html

http://www.utpb.edu/scimath/wkfield/mod3/Theorem.htm