How to find the electric field coming from a sphere WITHOUT using Gauss' law?

[anban]

Homework Statement

How do I find the electric field at a point above the center of a charged sphere? Assume the sphere is a shell.

Homework Equations

The Attempt at a Solution

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I know there will only be a z component to the electric field, because x and y components will cancel by symmetry. I think the process will have to involve integrating over the surface of the sphere. Where do I start?

More things I know (or think I know):
dq = σdA
The dA terms will point radially from the sphere.

 

[mfb]

I think the process will have to involve integrating over the surface of the sphere. Where do I start?

With a coordinate system, and with an expression for the radial component of the electric field as function of the distance to your charge.
You can use symmetry to reduce the two-dimensional integral to a one-dimensional integral quickly.

 

[Anonymous2258]

Can you use shell theorem to just treat it as a point charge, as long as the charge is symmetrically distributed?

 

[mfb]

That would be the result of Gauß' law ;).

 

[Nickie]

To find the electric field at a point above the center of a charged sphere, without using Gauss' law, we can use the principle of superposition. This principle states that the total electric field at a point is the vector sum of the individual electric fields produced by each charge.

First, we can divide the sphere into infinitesimal rings or shells, with each shell having a small charge element dq. The electric field at a point P, above the center of the sphere, can be found by summing up the electric fields produced by each shell.

To find the electric field produced by each shell, we can use Coulomb's law, which states that the electric field produced by a point charge is given by E = kq/r^2, where k is the Coulomb's constant, q is the charge, and r is the distance between the charge and the point P.

Since the charge elements on each shell are distributed uniformly, we can consider the shell as a point charge located at the center of the shell. This simplifies the calculation of the electric field produced by each shell.

We can then use the principle of superposition to sum up the electric fields produced by each shell to find the total electric field at point P. Since the electric field is a vector quantity, we must consider the direction of each electric field produced by each shell.

Since the electric field produced by each shell will have a component in the z direction, we can use the symmetry of the problem to simplify the calculation. We can consider the electric field produced by a single shell as a point charge located at the center of the sphere, and the distance between this point charge and point P will be the distance between the center of the sphere and point P.

We can then use the formula E = kq/r^2 to calculate the electric field produced by each shell and sum them up to find the total electric field at point P.

In summary, to find the electric field at a point above the center of a charged sphere without using Gauss' law, we can use the principle of superposition and consider the sphere as a collection of point charges located at the center of each shell. By summing up the electric fields produced by each shell, taking into account the direction and magnitude of each electric field, we can find the total electric field at point P.