Date: Sun, 30 Nov 2003 13:46:32 -0600
From: Delta <deltapsi@sympatico.ca>
Newsgroups: sci.physics
Subject: electric potential and change in potential
Can someone explain the mathematics of the equations for electric potential
and change in potential?
If you are considering electrostatics, then two of maxwell's equations are:
[Where: epsilon nought has been set to 1, 'E' is the electric
field vector, 'Del' is the gradient operator vector, 'rho' is the charge
density, '.' means dot product, and 'X' means cross product... whew.]
1) Del.E = rho
2) Del X E = 0
Because of equation 2 you can write E as the gradient of a scalar function,
called "the electric potential" (I'll denote it with V).
So Let E = - Del(V) [the minus sign is conventional]
plug that into equation 1 and get:
-Del^2(V) = rho
Then, typically in electrostatics problems, you are give rho and asked to
solve for V. One typical problem is where rho is the charge distribution
for a point charge...i.e. zero everywhere except right where the charge
is. To solve a problem like this one important mathematical method is
"Green's function solution". For example... say there is a single charge
located at the origin. Our equation becomes:
-Del^2(V(x)) = delta(x)
Here the solution, V(x) _is_ the green's function (that goes to zero as
x goes to infinity... but anyway... the best way to solve this is to
fourier transform (another good mathematical technique to know):
Fourier transform to get:
k^2 V(k) = 1
which means
V(x) = Intrgral[Exp[ikx](1/k^2),{k,-Infinity,Infinity}]
do the integral and get (for example in 3-D):
V(x) = 1/(4 Pi |x|)
To summarize, some of the mathematics I used to solve these equations
were: vectors, calculus, green's functions, fourier transforms, etc.
adam
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