# How to Solve Laplace's Equation for a Point Charge Near a Grounded Plane?

• RickRazor
In summary, the conversation discusses finding the potential at different points in space for a point charge situated at a distance d from a grounded conducting plane. The speaker wants to solve this problem without using the image charge idea and assumes azimuthal symmetry and takes the zonal harmonics. The potential is given by a series of terms involving coefficients A1, C1, C2, C3, etc. The potential vanishes when cos(theta) = d/r, and a condition is derived to progress further. The speaker mentions that solving this without images would be involved and suggests using Cartesian or cylindrical coordinates. They also mention solving it using integral transform techniques and contour integration.
RickRazor
Moved from technical forum so the template is missing
A point charge q is situated at a distance d from a grounded conducting plane of infinite extent. Find the potential at different points in space.

I want to solve this problem without using the image charge idea.

I assumed azimuthal symmetry and took the zonal harmonics. And we know that as r tends to infinite the potential in the opposite side of the conducting plane goes to a constant. So we are left with the Pn(theta)r^{-(n+1)} terms.

$$V(r,\theta)=A_1 + \frac{C_1}{r} + \frac{C_2}{r^2}\cos(\theta)) + \frac{C_3}{2}(3\cos^2(\theta) - 1) + ...$$

When cos(theta)=d/r, the potential vanishes. So we get the condition:

$$A_1 + \frac{C_1}{r} + \frac{C_2}{r^2}\frac{d}{r} + \frac{C_3}{2}(3\frac{d^2}{r^2} - 1) + ... =0$$

for all r

Now how to progress further? How do I get the coefficients

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I don't think you are answering the same question in the image - it just asks for the total charge on the plane (which you can compute from the image solution). What is the real question you are trying to answer?

Jason

By the way, if you want to solve without images, you will have better luck I think if you use Cartesian or cylindrical coordinates since your boundary is a plane. It will be fairly involved - more difficult than the kinds of problems in a typical undergrad (in US) textbook like Griffiths. For fun I solved it using integral transform techniques and contour integration, but other approaches could be used. I'm pretty sure that whatever way you solve it, instead of a discrete sum of functions you will find a continuous sum (an integral). EDIT: the integral can be evaluated to yield the image solution, of course.

Jason

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RickRazor

## 1. What is Laplace's Equation?

Laplace's Equation is a second-order partial differential equation that is used to describe the behavior of potential fields, such as gravitational or electrostatic fields. It is named after the French mathematician and astronomer, Pierre-Simon Laplace.

## 2. Why is Laplace's Equation important?

Laplace's Equation is important because it is a fundamental equation in many areas of physics, including electromagnetism, fluid dynamics, and heat transfer. It allows us to mathematically model and predict the behavior of these physical phenomena.

## 3. What is the solution to Laplace's Equation?

The solution to Laplace's Equation is a function that satisfies the equation and its boundary conditions. In most cases, it is not possible to find a general solution, so specific techniques must be used to solve it for different scenarios and boundary conditions.

## 4. How is Laplace's Equation solved?

Laplace's Equation can be solved using various techniques, such as separation of variables, Fourier series, or numerical methods. The choice of method depends on the specific problem and boundary conditions.

## 5. What are some applications of Laplace's Equation?

Some common applications of Laplace's Equation include the calculation of electric potential in electrostatics, the prediction of temperature distribution in heat transfer problems, and the modeling of fluid flow in aerodynamics. It is also used in image processing and pattern recognition.

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