# One E field eq in/outside sphere of charge?

• cefarix
In summary, the field inside and outside of a sphere of charge depends on the distance from the center, but is undefined at the surface.

#### cefarix

What is the single equation that describes the E field both inside, outside, and at the surface of a sphere of charge? At the hyperphysics website they give two different equations for both situations, the outside eq is inverse square, and the inside equation is direct linear.

What do you mean by "single equation"? As hyperphysics correctly states, the field within the surface depends linearly on the distance from the center, but outside the surface it depends on the distance squared. That is the formula describing the field at any point.

The equations are two though, one for the field inside, and separate one for the field outside. I suppose there is no way to directly combine these two equations. However can't we derive an equation which, I guess would be in integral form, summing up the flux through infinitesimal patches over the sphere's surface?

The one equation you appear to be refeering to is the first maxwell equation

$$\vec{\nabla}\cdot \vec{E} = \frac{\rho}{\epsilon_0}$$

or

$$\int \vec{E}\cdot d\vec{a} = \frac{Q_{int}}{\epsilon_0}$$

in integral form.

When we solve it for a charged sphere though, it turns out that E is of a different form inside and outside and is undefined directly at the surface.

cefarix said:
The equations are two though, one for the field inside, and separate one for the field outside. I suppose there is no way to directly combine these two equations. However can't we derive an equation which, I guess would be in integral form, summing up the flux through infinitesimal patches over the sphere's surface?
There is just one equation. However, because the gradient is discontinuous across the surface, when we write the equation as an analytic function, it must be split into two regimes. If this bothers you, realize that if you were to try to find a single continuous function to describe the density rho both inside and outside the sphere, you would end up with the same problem: no way to do it.

## 1. What is the formula for the electric field inside a charged sphere?

The formula for the electric field inside a charged sphere is E = kQr/r^3, where E is the electric field, k is the Coulomb's constant, Q is the charge of the sphere, and r is the distance from the center of the sphere.

## 2. How does the electric field inside a charged sphere vary with distance from the center?

The electric field inside a charged sphere varies inversely with the distance from the center. This means that as the distance from the center increases, the electric field decreases.

## 3. What is the relationship between the electric field inside and outside a charged sphere?

The electric field inside a charged sphere is constant and directed towards the center, while the electric field outside the sphere follows the inverse square law and is directed away from the center.

## 4. Can the electric field inside a charged sphere be negative?

No, the electric field inside a charged sphere cannot be negative. This is because the electric field is determined by the charge of the sphere, which is always positive.

## 5. How does the electric field inside a charged sphere differ from that of a point charge?

The electric field inside a charged sphere is constant and directed towards the center, while the electric field of a point charge varies with distance and is directed away from the charge. Additionally, the electric field inside a charged sphere is only present within the sphere, while the electric field of a point charge extends to infinity.