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Old May 20th 2017, 12:46 PM   #1
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Gauss's law using a cube

I thought I would try and verify Gauss's law for a cube of sides length a, centred at the origin, just for fun.

I didn't include the cross-product term in the surface integral as far as I can tell it equals unity anyway. The 3/2 power term comes from combining with the vector projection of the flux onto the area vector of the cube. The projection factor was as follows:



I came up with this surface integral.



Using wolfram alpha I got this to evaluate to:



This is for one face of the cube so need to multiply that by 6 for each face. However Gauss's law is supposed to be independent of shape and here it seems dependent upon the size of the box!! I have a feeling I may be mixing up surface and volume integrals. Am I missing an obvious surface parameterization here? However surface integrals always seem to be double and if anything this should be simpler than a sphere (which I worked through previously).

I should add that I evaluated this integral between 0 and a/2 and doubled the result rather than integrating between -a/2 and a/2 because the latter always evaluated to zero. The problem seems to be any integral of an even function evaluates to zero around equal and opposite upper and lower bounds. I know it shouldn't be zero so I did the former which probably means I set this problem up incorrectly but I can't quite see why.

This is the answer I was expecting:




Can anyone see any obvious mistakes I have made?

Last edited by kiwiheretic; May 20th 2017 at 04:49 PM. Reason: Clean up last eqn
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Old May 21st 2017, 06:24 AM   #2
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You say
I should add that I evaluated this integral between 0 and a/2 and doubled the result rather than integrating between -a/2 and a/2
but the integral you show is from -1/a to 1/a. Where did that come from?

The problem seems to be any integral of an even function evaluates to zero around equal and opposite upper and lower bounds.
No, it isn't. The integral of any odd function is even, F(-a)= F(a), so that F(a)- F(-a)= F(a)- F(a)= 0. Then integral of any even function is odd, F(-a)= -F(a), so that F(a)- F(-a)= F(a)+ F(a)= 2F(a).
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Old May 21st 2017, 01:53 PM   #3
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Originally Posted by HallsofIvy View Post
You say

but the integral you show is from -1/a to 1/a. Where did that come from?
The cube is centered at the origin with vertices at co-ordinates (1/a,1/a,1/a), (1/a,1/a,-1/a), (1/a,-1/a,1/a), (1/a,-1/a,-1/a), (-1/a,1/a,1/a), (-1/a,1/a,-1/a), (-1/a,-1/a,1/a), (-1/a,-1/a,-1/a).

I tried to cheat by calculating one face of the cube only thinking to multiply by 6 afterwards. (cube has 6 faces). I also tried to cheat by dividing a face into four quadrants, ie (x positive, y positive), (x, negative, y positive), (x positive, y negative) and (x negative, y negative). I chose to evaluate with 0 < x < 1/a and 0 < y < 1/a for both positive thinking to multiply the result by 4 as there are 4 quadrants. In other words using symmetry arguments where ever possible.

No, it isn't. The integral of any odd function is even, F(-a)= F(a), so that F(a)- F(-a)= F(a)- F(a)= 0. Then integral of any even function is odd, F(-a)= -F(a), so that F(a)- F(-a)= F(a)+ F(a)= 2F(a).
Well, if we use an even function like y = x^2 as an example and evaluate it around -1 < x < 1 then I(Y(x)) = x^3/3 where I(Y(1)) - I(y(-1)) = 1/3 - - 1/3 = 2/3. Ohhh, oops. That's what happens when I try to do this too late at night to battle insomnia. ;-)

I tried again but leaving out the constant factor



for brevity.



Then integrating with respect to y


substituting x=1/a for the result and doubling it this time because when we substitute for x = -1/a we are same but for a sign but then we also subtract the two.




substituting in x = 1/a yields



and x = -1/a yields



yielding



which is where I think SymPy Live led me up the garden path by telling me it was zero.

Doing the same for y = 1/a



and y = -1/a



this time yielding



Ok, that's a lot better than what I had but we still get the total flux is dependent on the area of the sides of the cube and not just the charge inside according to what Gauss's law says so what am I missing here?

Last edited by kiwiheretic; May 21st 2017 at 03:02 PM.
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Old May 21st 2017, 03:31 PM   #4
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Ooops, my cube corners are wrong, should be (a/2,a/2,a/2), (a/2,a/2,-a/2),(a/2,-a/2,a/2),(a/2,-a/2,-a/2),(-a/2,a/2,a/2),(-a/2,a/2,-a/2),(-a/2,-a/2,a/2),(-a/2,-a/2,-a/2)

Now when I substitute in integration limits -a/2 < x < a/2 and -a/2 < y < a/2 I get:



and that "a" term cancels out with the factor in



Ok, looks like I was half asleep when I was trying to do this last time. Looks like I solved it on my own (with some prompting from HallsofIvy). :-)
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Old May 21st 2017, 03:33 PM   #5
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I wonder what that "tan" expression represents in this case.
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