Laplace Equation for Cylinder
Instead of a rectangular prism, consider using separation of variables to find the potential inside a cylinder
a. What coordinate system would you use? Explain.
b. Write the Laplacian in this coordinate system.
c. What general form of function would you assume V in this case? Explain.
d. Apply the Laplacian you wrote in part b. to this function.
e. Can you manipulate your expression so each term only depends on one variable? If not, can you get one or two variables to only be in a single term?
f. Write the associated ordinary differential equations for the variables you separated above.
g. What is the Laplacian now, after you’ve separated out the variables you can? (You can substitute your equation for part f into the partial differential equation to eliminate the derivatives of that variable).
One should use a coordinate system that has the same symmetry as the geometry give. Hence the cylindrical coordinate system $(r,θ,z)$ is required.
The general Laplace equation for potential V is ($ρ$ is local density of charge)
For simplicity in the following we will consider a constant charge distribution (and also zero) so that in cylindrical coordinates Laplace equation becomes:
$(∇^2 V(r,θ,z)=) (1/r)*∂/∂r (r*∂V/∂r)+(1/r^2) *(∂^2 V)/(∂θ^2 )+∂V/∂z=0$
The general form of V that satisfies the above equation is
Where R is the radial part, M is the angular part and Z is the axial part.
Applying the above Laplacian to the chosen V one can separate the variables:
$(MZ/r)*(∂R/dr)+MZ*(∂^2 R/∂r^2 )+(RZ/r^2) *(∂^2 M/∂θ^2 )+RM*(∂^2 Z/∂z^2 )=0$
Or divided with V= RMZ
$(1/Rr)*(∂R/dr)+(1/R)*(∂^2 R/∂r^2 )+(RZ/r^2) *(∂^2 M/∂θ^2 )+(1/Z)*(∂^2 Z/∂z^2 )=0$
$(1/Rr)*(∂R/dr)+(1/R)*(∂^2 R/∂r^2 )+(1/Mr)^2 *(∂^2 M/∂θ^2 )=-(1/Z)*(∂^2 Z/∂z^2 )$
Since on the right there is only z variable and Z part of function on the left side the remaining other two variables and parts, this equation can be true only if both left and right terms are equal to a constant.
$(-1/Z)*(∂^2 Z/∂z^2)=-∝^2$ (constant) that is $(∂^2 Z/∂Z^2 )-∝^2*Z^2=0$ (2nd order homogeneous diff.eq)
The left side is:
$(r/R)*∂/∂r (r*∂R/∂r)+∝^2*r^2=-(1/M) (∂^2 M)/(∂θ^2 )$
which again is true just if both sides are equal to a constant
$-(1/M)*(∂^2 M/∂θ^2 )=+n^2$ (constant) that is $(∂^2 M/∂θ^2 )+n^2*M^2=0$
$r*(∂/∂r) (r*∂R/∂r)+R(∝^2*r^2-n^2 )=0$ which is known as a Bessel equation
Back into the original Laplace equation one has:
$(R/r^2) *(∝^2*r^2-n^2 )-n^2/r^2 +∝^2=0$