# What Is the Electric Field at Various Distances From a Charged Spherical Shell?

• 22steve
In summary, the problem involves a spherical shell with uniform volume charge density and different radial distances. The electric field at these distances is calculated using the equations for charge, flux, and electric field. The resulting values are 18094, 82, and 1813 when r = 1.5, r = b, and r = 3b respectively. The charge inside the sphere at r = 1.5 m is the same as the charge enclosed for r = b and r = 3b.
22steve

## Homework Statement

The figure below shows a spherical shell with uniform volume charge density ρ = 1.63 nC/m3, inner radius a = 6.0 cm, and outer radius b = 2.70a. What is the magnitude of the electric field at the following radial distances:
r=1.5, r=b, r=3b

## Homework Equations

Q= sigma dV
flux = E x A
flux = charge enclosed/permittivity constant (8.85e-12)
k x q/r^2 = E (electric field outside a spherical shell of charge)

E = k x r(q/R^3) R= radius r= distance from center of shell to point where E is measured.

## The Attempt at a Solution

I've tried integrating, and substituting different values of q into the Electric Field equations. I've gotten 18094, 82, and 1813 as answers. I would show my integration techniques, but it's kinda hard to do on the keyboard. haha

Part I:
When r = 1.5, what's charge inside the sphere of r = 1.5 m?

When r = b or r =3b, that sphere encloses all the charge. So, Q is same but Area changes for part II and part III

I would first like to commend you for attempting to solve the problem using equations and calculations. However, there are a few points that need to be addressed.

Firstly, in your equations, you have used the symbol "sigma" which typically represents surface charge density, but the problem states that the charge density is uniform throughout the volume of the spherical shell. Therefore, we should use the symbol "rho" instead.

Secondly, it seems that you have tried to use different equations for finding the electric field at different radial distances. However, for a spherical shell with uniform charge density, the electric field at any point outside the shell is given by E = kq/r^2, where q is the total charge enclosed within the Gaussian surface of radius r.

Finally, when solving for the electric field, it is important to consider the direction as well. Since the problem does not specify the sign of the charge, we cannot determine the direction of the electric field. Therefore, we can only solve for the magnitude of the electric field at each radial distance.

To solve this problem, we can use Gauss's Law, which states that the electric flux through a closed surface is equal to the charge enclosed divided by the permittivity constant. In this case, the closed surface can be taken as a spherical Gaussian surface with radius r.

At r = 1.5, the charge enclosed within the Gaussian surface is 0 since it is outside the spherical shell. Therefore, the electric field is 0.

At r = b, the charge enclosed within the Gaussian surface is the total charge of the spherical shell, which is given by q = rho x (4/3)π(b^3 - a^3). Substituting the values given in the problem, we get q = 1.63 x (4/3)π[(2.70a)^3 - (6.0 cm)^3] = 1.63 x 0.001 x 4π(2.70^3 - 6.0^3) = 1.63 x 0.001 x 4π(19.683) = 0.1285 nC. Therefore, the electric field at r = b is given by E = kq/r^2 = (8.99 x 10^9)(0.1285 x 10^-9)/(2.70 x 0.06)^2 = 0.

## 1. What is a spherical uniform charge?

A spherical uniform charge is a type of charge distribution where the electric charge is evenly distributed on the surface of a sphere. This means that the charge density is the same at every point on the sphere.

## 2. How is a spherical uniform charge different from a point charge?

A point charge is a theoretical concept where all the electric charge is concentrated at a single point. On the other hand, a spherical uniform charge has its charge evenly distributed on the surface of a sphere, making it a more realistic representation of a physical object.

## 3. How is the electric field of a spherical uniform charge calculated?

The electric field of a spherical uniform charge can be calculated using Coulomb's law, which states that the electric field at a point is directly proportional to the magnitude of the charge and inversely proportional to the square of the distance from the charge.

## 4. Can a spherical uniform charge have a net charge of zero?

Yes, a spherical uniform charge can have a net charge of zero if the positive and negative charges are evenly distributed on the surface of the sphere. In this case, the electric field outside the sphere will be zero.

## 5. What are some real-life examples of spherical uniform charges?

Spherical objects such as metal balls, balloons, and planets can be considered as examples of spherical uniform charges as their electric charge is evenly distributed on their surfaces. Additionally, electrons in an atom can also be treated as a spherical uniform charge due to their even distribution around the nucleus.

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