# Electrical field outside spherical shell

• glederfein
In summary, the problem involves an insulated spherical shell with a surface charge density of \sigma(\theta)=\sigma_0\sin(\theta) when \theta is the angle from the z axis. The task is to calculate the electric field outside the shell at z=R using Gauss's Law. The solution involves calculating the overall charge of the shell and using Gauss's Law to find the electric field at the specific point.

## Homework Statement

An insulated spherical shell with its center in the origin and radius R has a surface charge density of $$\sigma(\theta)=\sigma_0\sin(\theta)$$ when $$\theta$$ is the angle from the z axis. Calculate the electrical field outside the shell at point z=R.

## Homework Equations

Gauss's Law
$$\int\int{\vec{E}\cdot\vec{dS}}=4\pi{Q_{in}}$$

## The Attempt at a Solution

I tried first calculating the overall charge of the spherical shell:
$$Q=R^2\int_0^{2\pi}\sigma_0\sin^2\theta{d\theta}\int_0^\pi{d\phi}=\frac{R^2\sigma_0}{2}\int_0^{2\pi}(1-\cos{2\theta})d\theta\cdot\pi=\frac{1}{2}\pi{R^2}\sigma_0\cdot{2\pi}=\pi^2R^2\sigma_0$$
However, I don't see how Gauss's Law can help me find the electrical field at that specific point.

anyone?

Perhaps I should somehow calculate $$\int{\frac{dq}{r^2}}$$ ?

Your attempt at calculating the overall charge of the spherical shell is correct. However, Gauss's Law can indeed help you find the electrical field at a specific point outside the shell.

Gauss's Law states that the electric flux through a closed surface is equal to the enclosed charge divided by the permittivity of the medium. In this case, we can consider a Gaussian surface in the shape of a sphere with radius r>R, centered at the origin.

Using the symmetry of the problem, we can see that the electric field is uniform and radial, pointing away from the center of the sphere. Hence, the electric field at point z=R will be equal to the electric field at the surface of the sphere, which is given by the electric flux through the Gaussian surface divided by the surface area of the sphere.

We can calculate the electric flux through the Gaussian surface by using the fact that the electric field is perpendicular to the surface at every point. Thus, we have:

\int\int{\vec{E}\cdot\vec{dS}}=E\cdot4\pi{r^2}

And since the enclosed charge is equal to the overall charge of the spherical shell, we have:

\int\int{\vec{E}\cdot\vec{dS}}=\frac{\pi^2R^2\sigma_0}{\epsilon_0}

Equating the two equations and solving for E, we get:

E=\frac{\pi\sigma_0R}{2\epsilon_0}

Therefore, the electric field at point z=R is given by E=\frac{\pi\sigma_0R}{2\epsilon_0}, where \epsilon_0 is the permittivity of the medium.

## 1. What is an electrical field?

An electrical field is a physical quantity that describes the force experienced by a charged particle in the presence of other charged particles or electrically charged objects. It is represented by electric field lines and is measured in units of volts per meter (V/m).

## 2. How is the electrical field outside a spherical shell calculated?

The electrical field outside a spherical shell is calculated using Coulomb's Law, which states that the electric field at a certain point is directly proportional to the magnitude of the charge and inversely proportional to the distance from the charge. It can be calculated by dividing the magnitude of the charge by the square of the distance from the center of the spherical shell.

## 3. What is the direction of the electrical field outside a spherical shell?

The direction of the electrical field outside a spherical shell is always perpendicular to the surface of the shell. This means that the electric field lines will point away from the surface of the shell if the charge is positive and towards the surface of the shell if the charge is negative.

## 4. How does the electrical field outside a spherical shell change with distance?

The electrical field outside a spherical shell follows an inverse square law, meaning that as the distance from the shell increases, the electrical field strength decreases. This can be seen in the equation for the electric field, where the field strength is inversely proportional to the square of the distance from the center of the shell.

## 5. Can an electrical field exist inside a spherical shell?

No, an electrical field cannot exist inside a spherical shell. This is because the charge inside the shell would cancel out the electric field, resulting in a net electric field of zero. Therefore, any charged particles inside the shell would experience no force from the electric field.