I am trying to solve the following: $$\vec \nabla ^2\Phi=\delta(x-x_0)\delta (y-y_0)$$ where $\Phi$ is a scalar function of the form $\Phi(x,y)$. Boundary conditions are $\Phi(0,y)=\Phi(x,0)=0$. The exercise is to find the potential in the area $x>0,y>0$ that arises due to two infinite electric grounded planes lying perpendicular one the other on the $x-z$ and the $y-z$ planes, with the following charge distribution: $$\rho(\vec r)=\frac{q}{l} \frac{\text{Coulomb}}{\text{meter}}\delta(x-x_0)\delta(y-y_0)$$ (A charged wire parallel to the z axis)

How do I find $\Phi$?

Green's function in 2-d is $\frac{1}{2\pi} \ln r$, but is it a 2-d problem or 3-d?

Once choosing the correct function, how is the solution carried out? Should I solve for $$\vec \nabla ^2G(x,y,x_0,y_0)=0$$or for

$$\vec \nabla ^2G(x,y,x_0,y_0)=\frac{\rho(x,y)}{\epsilon_0}?$$ Any hint would be helpful.

  • $\begingroup$ Is $\Phi$ a function $\Phi(x,y)$ or is it $\Phi(x,y,z)$? In the first case you have a point charge, in the second case you have a line charge. The situation is different between the two. $\endgroup$ – Ian Aug 10 '16 at 15:44
  • $\begingroup$ It has no dependence on $z$ because of the geometry of the later described setup. $\endgroup$ – E Be Aug 10 '16 at 15:45
  • $\begingroup$ I understand that, but that comes from the solution, it's not intrinsic. Intrinsically $\Phi$ depends on all your spatial variables, of which there are three. $\endgroup$ – Ian Aug 10 '16 at 16:11
  • $\begingroup$ I agree with you. $\endgroup$ – E Be Aug 10 '16 at 16:15
  • $\begingroup$ In that case, the relevant Green's function is 3D, i.e. the solution to $\nabla^2 \Phi = \delta(x) \delta(y) \delta(z)$. $\endgroup$ – Ian Aug 10 '16 at 16:22

If you have an infinite wire, you should try to use cilindrical coordinates insted of cartesian so the solution is $1/\rho$ where $\rho^2=x^2+y^2$. Then you should think an appropiated cutoff, like multiplying by an appropiated Heaviside function or stuff like that. The book of Jackson-Classical Electrodynamics has a lot of examples and a good mathematical and physical approach


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.