Homework2 - Homework 2 Answers, 95.657 Electromagnetic...

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Homework 2 Answers, 95.657 Electromagnetic Theory I Dr. Christopher S. Baird, UMass Lowell Jackson 1.5 The time-averaged potential of a neutral hydrogen atom is given by = q 4  0 e − r r 1 1 2 r where q is the magnitude of the electronic charge, and 1 = a 0 / 2 , a 0 being the Bohr radius. Find the distribution of charge (both continuous and discrete) that will give this potential and interpret your result physically. SOLUTION: The Poisson equation links charge densities and the electric scalar potential that they create. We use it here to find the charge density: 2 =− 0 Expand this in spherical coordinates: 1 r 2 r r 2 ∂ r 1 r 2 sin ∂ sin ∂ ∂ 1 r 2 sin 2 2 ∂ 2 =− 0 The potential is spherically symmetric, so that the potential depends only on the radial coordinate. The partial derivatives of the potential are all zero, except for with respect to the radial component. 1 r 2 r r 2 ∂ r =− 0 Evaluate the equation explicitly: 1 r 2 r r 2 r [ q 4  0 e − r r 1 1 2 r ] =− 0 q 4  0 1 r 2 r r 2 r [ e − r r ] r 2 r [ 2 e − r ] =− 0 q 4  0 1 r 2 r r 2 [ 1 r r e − r e − r r 1 r ] 2 2 r 2 e − r =− 0 q 4  0 1 r 2 r − r e − r r 2 e − r r 1 r 2 2 r 2 e − r =− 0
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q 4  0 − r 2 e − r 3 2 e − r q 4  0 1 r 2 r r 2 e − r r 1 r =− 0 = q 3 8 e − r e − r q 4 1 r 2 r r 2 r 1 r Now we must be careful because 1/ r blows up at the origin. Split the last term into two cases: = q 3 8 e − r e − r q 4 1 r 2 r r 2 r 1 r if r ≈ 0 = q 3 8 e − r e − r q 4 1 r 2 r r 2 r 1 r if r > 0 Away from the origin, 1/ r does not blow up and the derivatives can be evaluated normally. The last term ends up equating to zero, so that our equations now becomes: = q 3 8 e − r e − r q 4 1 r 2 r r 2 r 1 r if r ≈ 0 = q 3 8 e − r if r > 0 At r ≈ 0, we have e − r = 1 so that the two cases become: = q 3 8 q 4 1 r 2 r r 2 r 1 r if r ≈ 0 = q 3 8 e − r if r > 0 Now use the relation: 2 1 r =− 4  r which when evaluated explicitly becomes: 1 r 2 r r 2 r 1 r =− 4  r Plug this into the above set of equations:
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=− q 3 8 q  r if r ≈ 0 =− q 3 8 e − r if r > 0
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Homework2 - Homework 2 Answers, 95.657 Electromagnetic...

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