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Old May 9th 2016, 05:54 AM   #1
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Use of equations in electromagnetism question

I am having some trouble as to where the following equations came from when working out the following problems.

Problem1: A electron is held 5 nm above a conducting metal surface. Use the method of images to calculate the induced surface charge density immediately below the electron on the surface of the plate.

Equation used: sigma = -aQ/(2*pi(x^2+y^2+a^2)^3/2)

I can do the problem and I understand the solution but I just don't know where the equation came from.

Problem2: A electron is held 5 nm above a conducting metal surface. Use the method of images to calculate the force between the charge and the plate. Is the force attractive or repulsive?

Equation used: F = -Q^2/(16*pi*e0*a^2)

My question here is where did the 16pi come from? I though it was 4pi.

Any help at all would be greatly appreciated
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Old May 10th 2016, 12:01 PM   #2
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The first equation comes from a calculation of the potential V as a function of position z, and then taking the derivative of epsilon_0 V(z) with respect to z and setting z=0. Here's a description of that: https://en.wikipedia.org/wiki/Method_of_image_charges

As for the second equation - you're right that the equation for force has a factor of 4, but here the distance between the two particles is 2a, so that distance squared becomes 4 a^2. Hence the equation ends up with a factor of 16.

Last edited by ChipB; May 10th 2016 at 12:20 PM.
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Old May 10th 2016, 12:13 PM   #3
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As always Chip, you've been awesome.

Thank you so much for your help.
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