# When to use Laplace's or Poisson's equation for calculating potential

• mdwerner
In summary, the conversation is about understanding when to use Laplace's equation versus Poisson's equation. In the given example of a rectangular pipe with specified potential, Laplace's equation is used because there are no charges inside the pipe. This means that Laplace's equation is a special case when there are no charges present.
mdwerner

## Homework Statement

The problem that I'm having is simply understanding the difference in when to use the laplace's equation vs the poisson's equation. Here is an example of a question : "A rectangular pipe, running parallel to the z-axis (from -∞ to + ∞), has three grounded metal sides, at y = 0, y = a, and x = 0. The fourth side, at x = b, is maintained at a specified potential V(y)."
I can see in the solution that we are to use Laplace's equation and not Poisson's equation. Why? What determines this? The solution does not state this, it simply states that it IS.
Any help is much appreciated.

## Homework Equations

∇^2 V = 0 (Laplace's Equation)
∇^2 V = - ρ / ε (Poisson's Equation)

## The Attempt at a Solution

I see that Laplace's equation would be a special case in which the region we are talking about is devoid of any charges - In the example we have a conducting material with a potential in one region that isn't present in another. Does this not mean that there are charges present? If charges are present, then Poisson's Equation must be used. Any help clarifying this would mean a lot - thanks.

Last edited:
You're presumably trying to find the potential inside the pipe. Since there are no charges inside the pipe, you use Laplace's equation.

## 1. What is the difference between Laplace's and Poisson's equation?

Laplace's and Poisson's equations are both mathematical equations used to calculate the potential (or electric field) in a given region. The main difference between the two is that Laplace's equation assumes that the charge distribution is zero, while Poisson's equation allows for non-zero charge distributions.

## 2. When should I use Laplace's equation?

Laplace's equation is most commonly used when the charge distribution in a given region is known to be zero. This is often the case when dealing with problems involving conductors, since charge tends to distribute evenly on the surface of a conductor.

## 3. When should I use Poisson's equation?

Poisson's equation is used when the charge distribution in a given region is not zero. This is often the case when dealing with problems involving dielectrics or other materials with non-zero charge distributions.

## 4. Can Laplace's and Poisson's equations be used interchangeably?

No, Laplace's and Poisson's equations cannot be used interchangeably. They are two distinct equations with different assumptions about the charge distribution in a given region. Using the wrong equation may result in incorrect calculations.

## 5. How do I know which equation to use for a specific problem?

To determine which equation to use, you must first consider the charge distribution in the given region. If the charge distribution is known to be zero, then Laplace's equation should be used. If the charge distribution is non-zero, then Poisson's equation should be used.

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