# Electric Potential: Electric charge surrounded by two spherical shells

• Bryon
In summary, the question involves a point charge surrounded by two conducting spherical shells with specified radii and charges. The goal is to find the potential difference between two points on the outer shell, using the equations for electric field and potential. The use of Gauss' Law and the concept of an equivalent point charge is necessary to solve the problem.
Bryon

## Homework Statement

A point charge q = -8 μC is surrounded by two thick, conducting spherical shells of inner and outer radii
a1 = 0.3 m, a2 = 0.4 m, a3 = 0.7 m, and a4 = 0.8 m respectively. The inner shell is uncharged; the outer
shell has a net charge Q = -10 μC. At this point in the problem, the potential at infinity is unspecified.

(c) V(a2) - V(a3) = ____________________ V

## Homework Equations

E = kQ/r^2

Guass’ Law: ϕ= ʃE·dA

V0 = - ʃE·dl

HELP: Identify the equivalent point-charge that gives the same electric fields everywhere in the region a2 < r < a3 as the specified charge distribution.
HELP: Use the electric potential function that corresponds to the equivalent point-charge problem.

## The Attempt at a Solution

Electric field of the sphere at a2:
E = k(-8uC)/(.4^2) = -449500 N/C

Im stuck. I figured out the first two problems but I am not sure how to relate the electric field to the potential, and I am not sure how to get the equivalent point charge. I assume I need to use guass' law to find the equivalent point charge but my queastion is do I just use the difference of a2 and a3 for the radius?

Last edited:
Assuming that the spherical shells are centered at the origin:

What is the electric field a distance r from the origin, where a2 < r < a3 ?

For a2 < r < a3:

$$V(r)-V(a_2)=-\int_{a_2}^{\,r}E(r)\,dr\ .$$

Ah. So I just integrate from 0.4 to 0.7? I feel stupid sometimes lol.

Va-Vb = - ʃ(kQ/r^2)dr = kQ((1/a)-(1/b))

Thanks for the help by the way!

## 1. What is electric potential?

Electric potential is the amount of electric potential energy per unit charge at a point in an electric field. It is measured in volts (V).

## 2. How is electric potential calculated?

The electric potential at a point is equal to the work done per unit charge in bringing a positive test charge from infinity to that point in an electric field. It can be calculated using the formula V = kQ/r, where k is the Coulomb's constant, Q is the charge, and r is the distance from the charge.

## 3. What is the significance of two spherical shells in electric potential?

Two spherical shells are often used to represent a system of electric charges, where one charge is surrounded by another charge. This allows for a simplified calculation of the electric potential at various points in the system.

## 4. How does the electric potential change between the two spherical shells?

The electric potential decreases as the distance from the inner shell increases, and then increases again as the distance approaches the outer shell. This is because the electric potential is directly proportional to the distance from the charge.

## 5. What factors affect the electric potential between the two spherical shells?

The electric potential is affected by the magnitudes of the charges on the shells, the distance between the shells, and the medium between the shells. It is also affected by any external charges or fields that may be present.

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