# A point charge in a sphere with V0

• rootX
In summary, solving a non-homogeneous PDE with a spherical point charge results in V being located at the center of the sphere.

#### rootX

If there is a point charge at the center of a hollow sphere that has voltage of V0. How would I go by solving the voltage and electric field distribution?

I know I can use Poisson (Laplace) to find distribution had there been no charge inside the sphere, and could use Gauss had there been no V on the sphere. But how about the combination?

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Hmmm. So there is a spherical shell with a voltage, and now we want to find the voltage inside the shell?

So it seems that what you'll have to do, unless I'm overlooking something, is

$$\nabla^2 V = \frac{q}{\epsilon_0}$$

You'll be solving a non-homogeneous PDE. It may be a tricky solution to work out. Of course, once you solve V, you can E from the gradient (which also may be not easy task).

The spherical symmetry makes this into a simple problem.

Since the point charge is placed at the very center of the sphere, that means that whatever charge redistribution it causes on the hollow sphere, retains the spherical symmetry. That means that the sphere provides no inwards E-field and acts as a point charge outwards (Remember that no charge has flowed off the sphere or onto it).

Use Gauss' Law and the capacitance of the sphere to solve for the charges, potentials and fields.

Mindscrape said:
Hmmm. So there is a spherical shell with a voltage, and now we want to find the voltage inside the shell?

So it seems that what you'll have to do, unless I'm overlooking something, is

$$\nabla^2 V = \frac{q}{\epsilon_0}$$

You'll be solving a non-homogeneous PDE. It may be a tricky solution to work out. Of course, once you solve V, you can E from the gradient (which also may be not easy task).

There is a spherical shell with a point charge Q at its center and I want to know how to find E/V distribution over all the space inside/outside.

$$\nabla^2 V = \frac{q}{\epsilon_0}$$

So the "q" will be just the charge that is located inside the sphere?
RoyalCat said:
The spherical symmetry makes this into a simple problem.

Since the point charge is placed at the very center of the sphere, that means that whatever charge redistribution it causes on the hollow sphere, retains the spherical symmetry. That means that the sphere provides no inwards E-field and acts as a point charge outwards (Remember that no charge has flowed off the sphere or onto it).

Use Gauss' Law and the capacitance of the sphere to solve for the charges, potentials and fields.

Because, there is a boundary condition that V0 is placed on the shell doesn't that affect the way Q induces charge over the shell or it can be just
Q/A

rootX said:
There is a spherical shell with a point charge Q at its center and I want to know how to find E/V distribution over all the space inside/outside.

$$\nabla^2 V = \frac{q}{\epsilon_0}$$

So the "q" will be just the charge that is located inside the sphere?

Right, this is just verbatim of Maxwell's Equation. It's been a while since I've solved a non-homogenous PDE, but it's just like for differential equations: the solution is the particular solution plus the homogeneous solution (the general solution is a well-known solution for this type of problem). Now generally, you'd use eigenfunction expansion to solve this, but I'm not sure how well that will work with Legendre polynomials. However, you should be able to just guess the particular solution. After all, you only need to figure out something that will give you a constant after two derivatives.

Again, I haven't yet worked it out myself, but maybe it won't be so bad now that I think about it. Let me know how it goes (or went if this is already past your due date).

## 1. What is a point charge in a sphere with V0?

A point charge in a sphere with V0 refers to a scenario in which a single electric charge is located at the center of a spherical object with a fixed electric potential, V0. The charge can be either positive or negative, and the electric potential represents the amount of work required to move a unit of charge from infinity to a specific location within the sphere.

## 2. How is the electric field affected by a point charge in a sphere with V0?

The electric field within the sphere is affected by the point charge in several ways. Firstly, the electric field at any point within the sphere is directly proportional to the charge, meaning that a larger charge will result in a stronger electric field. Secondly, the electric field is also affected by the electric potential, with a higher potential resulting in a stronger electric field. Additionally, the electric field is affected by the distance from the center of the sphere, with the strength decreasing as the distance increases.

## 3. What is the equation for the electric potential of a point charge in a sphere with V0?

The equation for the electric potential, V, at a point within the sphere is given by V = kQ/r, where k is the Coulomb's constant, Q is the charge of the point charge, and r is the distance from the center of the sphere.

## 4. How does the electric potential change as the distance from the center of the sphere increases?

As the distance from the center of the sphere increases, the electric potential decreases. This is because the electric potential is inversely proportional to the distance from the center, meaning that as the distance increases, the potential decreases.

## 5. What happens to the electric field and potential outside of the sphere?

Outside of the sphere, both the electric field and potential are affected by the presence of the sphere. The electric field is still inversely proportional to the distance from the center, but it also depends on the radius of the sphere. The electric potential, on the other hand, is constant outside of the sphere and is equal to V0, the potential at the surface of the sphere.