Sph Harmonics Homework: Find Potential for r>a, Contribution Using Superposition

In summary: My mistake. Sorry about misreading the phi boundary condition. In that case, you should still look for symmetry, but adding -V/2 to everything will not be useful.
  • #1
Faizan Samad
12
1

Homework Statement


A sphere of radius a has V = 0 everywhere except between 0 < θ < π/2 and 0 < φ < π. Write an expression in spherical harmonics for the potential for r > a. For which values of m are there contributions? Determine the contributions through l= 2. How could you determine the leading contribution using just superposition?

Homework Equations


Again I will be uploading images. I am only looking for answer in terms of a general formula. Not a fully drawn out calculation.
Screen Shot 2018-09-17 at 12.24.04 AM.png


The Attempt at a Solution


I will be uploading a solution. Again the solution they are looking for is very general. What I am confused about is that this problem implies that for certain values of m there are no contributions but that does not seem to be the case with these bounds of integration. Also I do not understand the superposition part of this question.
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  • #2
Look at the symmetry of the problem and compare it with the symmetry of the spherical harmonics.
 
  • #3
bobob said:
Look at the symmetry of the problem and compare it with the symmetry of the spherical harmonics.
I’m integrating all the phi terms they don’t seem to go to 0.
 
  • #4
Faizan Samad said:
I’m integrating all the phi terms they don’t seem to go to 0.

That's why you don't just throw math a problem that involves physics. It appears your potential independent of \phi, correct? If so,which m values should you not get zero answers for? Second, you were asked how you could guess a term or terms by superposition. Look at the symmetry of the Y_lm you can have after figuring out which ones those are (that you got by symmetry). You have the potential on a sphere which is V on the upper hemisphere and 0 on the lower hemisphere. It might help to shift the potential by -V/2 so that the symmetry is more obvious. Then you have a constant V/2 on the entire sphere and term and and a tem with +V/2 on the top hemisphere and -V/2 on the bottom hemisphere. When you superpose them you get V on the top half and zero on the bottom half. First thing you should always do in any problem is shift the origin or shift something to make as much symmetry explicit as possible. Then when you are done, you can shift back.
 
  • #5
bobob said:
It appears your potential independent of \phi, correct?
That's not correct. The potential is non-zero for only ##0 < \phi < \pi##.
 
  • #6
vela said:
That's not correct. The potential is non-zero for only ##0 < \phi < \pi##.
My mistake. Sorry about misreading the phi boundary condition. In that case, you should still look for symmetry, but adding -V/2 to everything will not be useful.
 
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Related to Sph Harmonics Homework: Find Potential for r>a, Contribution Using Superposition

1. What is the purpose of "Sph Harmonics Homework: Find Potential for r>a, Contribution Using Superposition"?

The purpose of this homework is to apply the concept of spherical harmonics to finding the potential for points outside of a given spherical region, or when r>a. It also involves using the principle of superposition to calculate the contribution of multiple sources to the overall potential.

2. What is the formula for calculating the potential for r>a?

The formula for finding the potential for points outside of a spherical region, or when r>a, is V(r,θ,φ) = Σ[ k_n * Y_n(θ,φ) * (a^n)/r^(n+1) ], where k_n is the multipole moment coefficient, Y_n(θ,φ) is the spherical harmonic function, a is the radius of the spherical region, and r is the distance from the center of the spherical region to the point of interest.

3. How do you use the principle of superposition to find the overall potential for multiple sources?

The principle of superposition states that the total potential at a point is equal to the sum of the individual potentials from each source. Therefore, to find the overall potential for multiple sources, you would first calculate the potential for each individual source using the formula mentioned in question 2. Then, you would add all of these individual potentials together to get the total potential at the point of interest.

4. What are some real-world applications of spherical harmonics and superposition?

Spherical harmonics and superposition have many practical applications, such as in physics, astronomy, and engineering. Some examples include calculating the gravitational potential of planets and stars, predicting the behavior of electromagnetic fields, and designing antennas for communication systems.

5. What are some common mistakes to avoid when working on this type of homework?

Some common mistakes to avoid when working on this type of homework include forgetting to account for the radius of the spherical region (a) in the formula, using the wrong values for the multipole moment coefficients, and not properly understanding the concept of superposition and how it applies to this problem. It is also important to double-check all calculations and units to avoid errors.

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