Charged Spherical Shell

Claimed by Chianne Connelly

The Main Idea

Charged objects create electric fields. Each object creates a different electric field depending on its shape, charge, and the distance to the observation location. A charged spherical shell acts like a point charge, so it uses the same equation as the electric field from a point charge.

A Mathematical Model

For an observation location outside of the sphere, the equation E_sphere = (1/4πε_0)(q/r^2)rhat should be used, where q is the charge of the object and r is the magnitude of the distance from the observation location to the source. However, if your observation location is inside of the sphere, E=0.

A Computational Model

(I spent a good amount of time trying to put images in this section but I could not manage to do so -- I'm sorry!)

If the observation location is outside of the shell, the electric field produced mirrors that of a point charge, due to the shape and charge distribution of the charged spherical shell. Say the shell is located at the origin, and the observation location is on the x-axis. The direction of the electric field produced by the shell at the observation location is in the x direction. This is because all of the other electric field vectors with y and x components cancel out in the y direction, leaving only the electric field in the x direction. The same logic would be used if the observation location was on any of the axes. For example, if the observation location had a unit vector of <1,1,0>, then the electric field would have components in the x and y directions, and their magnitudes would be whatever the value of the electric field was found to be multiplied by 1, since both the x and y components of the unit vector have values of 1.

If the observation location is anywhere inside of the spherical shell, then the electric field is zero. This is because all of the charges will cancel out.

Examples

Simple

A spherical shell of charge with a radius of 5 is located at the origin and is uniformly charged with q=+2. What is the electric field produced from the spherical shell at x=2?

E=0

Middling

A spherical shell of charge with a radius of 5m is located at the origin and is uniformly charged with q=+2e-7. What is the electric field produced from the spherical shell at x=10m?

E_sphere = (1/4πε_0)(q/r^2)rhat

E_sphere = (1/4πε_0)(2e-7/5^2)<1,0,0>

E_sphere = <72,0,0>

Difficult

A spherical shell of charge with a radius of 1m is located at the origin and is uniformly charged with q=+6e-8. What is the electric field produced from the spherical shell at (6,3,2)?

E_sphere = (1/4πε_0)(q/r^2)rhat

E_sphere = (1/4πε_0)(6e-8/7^2)<6/7,3/7,2/7>

E_sphere ≈ (11.02)<6/7,3/7,2/7>

E_sphere ≈ <9.45,4.72,3.15>

Connectedness

This topic is related to electric fields and the effects that electric fields can have on other objects. For example, electric fields can have effects on humans! The body's voltage can be increased, currents can be induced by the body, and electric charges can buildup on the surface of peoples' skin which is why they feel a tingling sensation when exposed to electric fields (such as from standing under a high voltage power line). This tingling is felt starting from voltages of 1,000 volts per meter. At that same voltage, there are microdischarges when a person touches something made of metal.

History

The electric field from a point charge was discovered by Charles Augustin de Coulomb, a French physicists. Coulomb's law was published in 1784. The law states that the electric field from a point charge is inversely proportional to the distance between the charged particle and the observation location. It also states that if the charge creating the electric field is positive, then the electric field will point radially outward. However, if the particle creating the field is negatively charged, then the electric field will point radially inward.

See also

Electric Force One application of electric fields due to point charges deals with finding electric force

Electric Field More general ideas about electric fields

Further reading

Principles of Electrodynamics by Melvin Schwartz ISBN: 9780486134673

External links

Internet resources on this topic

References

This section contains the the references you used while writing this page