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# 1.5.1-1.5.3

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## Richard Datwyler

on 26 September 2018

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#### Transcript of 1.5.1-1.5.3

1.5.1-1.5.3
Dirac Delta Function
1.5.1 Problems with Divergence
1.5.2 1D Dirac Delta Function
1.5.3 3D Dirac Delta Function
Review Practice
Take the divergence of
Now check it with the divergence theorem for a sphere of radius R.
0
problem
What did we do?

The divergence of is zero everywhere, but right at the origin.
But in the divergence theorem, there is no divide by zero, so we can feel pretty sure that its result is accurate.
Thus the volume integral of the diverged must result in the same value, and it must come from just the location at r = 0.
We see this all over, this is exactly what we will get from a point charge.
and it is also the same idea of a density of point particle.
1D Dirac Delta Function
It is not a true function, and we often view it as a limit of a function going to zero. It is more of a distribution function.
It is of most use when performing integrals, especially where the integral has a point location.(source or sink)
Uses
The dirac delta function is only 'used' inside an integral.
Evaluate the integral
D
C
B
A
3
-3
9
Practice
Evaluate the integral
D
C
B
A
3
-3
9
0
3D Delta Functions
Returning to the divergence of inverse square
We know that the integral over any volume is zero, save at the origin r = 0, where it evaluates as 4 pi.

This then leads to the divergence of the inverse square being the same as a delta function or:
Then in an integral
in application
Write volume charge density of a point charge q at r`.
check integral over all space
What is the volume charge density of a uniform, infinitesimally thin spherical shell of radius R and total charge Q.
to find Constant C perform integral of rho over all space and set equal to Q.
"I don't understand how equation 1.85 was set up and solved. "
'Three-dimensional delta function was like reading Chinese, can we please go over that?"
"Why is the dirac delta function is not well defined everywhere ?"
"May we go over problem 1.43(d.)? "
"Could we go over equation 1.100 in the textbook?"
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