Calculating E-field of Uniformly Charged Sphere

[Euclid]

I need to calculate the electric field both inside and outside a sphere with a charge density p. The trick is I am required to use Coulomb's Law. I cannot appeal to Gauss' Law.

My instructor suggests considering the electric field due to a ring of charge. Then to use that result to find the electric field of a spherical shell. Finally, I will use that result to find the electric field due to a sphere.

I've done the first part with no problems, and my answer checks with the book. (R=radius of ring, r=distance along axis of ring)
$$E=\frac{Qr}{4 \pi \epsilon_0 (r^2+R^2)^{3/2}}$$
However, I am stuck on how to use this result to find the E field due to a spherical shell.

I begin by considering a "ring element" of radius r and I consider a point on the x-axis (which happens to coincide with the axis of the ring). I need to find the dE associated with this ring element, but it's not so simple. It should be $$dE=\frac{dQ(d-x)}{4\pi\epsilon_0((d-x)^2+r^2)^{3/2}}$$ (d=x-coordinate of given point on x-axis, x=x-coordinate of plane through ring, r=ring radius) (this isn't showing up correctly: it's missing "dE" and "dQ") Here dQ, x, r are all changing and when I plug in the relations between them I get an ugly integral, that doesn't appear to come out with the 1/r^2 dependence. Any suggestions?

 

[anathema]

One approach you could take is to use the principle of superposition. This means that the electric field at a point due to a collection of charges is equal to the vector sum of the electric fields at that point due to each individual charge.

In the case of a spherical shell, you can think of it as a collection of many small rings of charge, each with a radius r and a charge density p. The electric field at a point along the x-axis due to one of these rings would be dE=\frac{dQ(x-d)}{4\pi\epsilon_0((x-d)^2+r^2)^{\frac{3}{2}}}.

To find the total electric field at this point, you would need to integrate over all the rings, taking into account the varying distance from the point to each ring. This would give you the expression:

E=\int_{0}^{2\pi}\int_{0}^{\pi}\frac{p(\sin\theta)(\rho^2\sin\theta)(\rho d\rho d\theta)(x-d)}{4\pi\epsilon_0((x-d)^2+\rho^2)^{\frac{3}{2}}}

where \rho is the distance from the origin to the ring, \theta is the angle between the x-axis and the line connecting the point to the ring, and p is the charge density of the shell.

This integral may look complicated, but it can be simplified by using some trigonometric identities and variable substitutions. In the end, you should get an expression that has the 1/r^2 dependence you are looking for.

It's also important to note that this approach assumes that the spherical shell has a finite thickness. If the shell is infinitely thin, then you would need to use a different method, such as finding the electric field due to the entire shell using Coulomb's Law and then taking the limit as the thickness approaches zero.

 

[M98Ranger]

Using Coulomb's Law to calculate the electric field due to a uniformly charged sphere can be a bit tricky, but it is certainly doable. As your instructor suggested, you can start by considering the electric field due to a ring of charge. This will help you find the electric field of a spherical shell, and then you can use the result to find the electric field of a sphere.

To find the electric field due to a ring, you correctly used Coulomb's Law and found the expression E=\frac{Qr}{4 \pi \epsilon_0 (r^2+R^2)^{3/2}}, where Q is the charge of the ring and r is the distance from the center of the ring to the point where you want to find the electric field. This result is correct and can be used to find the electric field of a spherical shell.

To find the electric field due to a spherical shell, you can divide the shell into infinitesimal rings and use the superposition principle to add up the contributions from each ring. This will give you an integral expression, but it should not be too complicated. You can also use the fact that the electric field inside a conductor is zero to simplify the integral. Once you have the electric field due to the shell, you can use this result to find the electric field due to a solid sphere by considering the shell as a collection of concentric shells with varying radii.

In summary, to find the electric field due to a uniformly charged sphere using Coulomb's Law, you can start by considering the electric field due to a ring of charge. Then use this result to find the electric field due to a spherical shell, and finally use this result to find the electric field due to a solid sphere. With careful consideration and application of the superposition principle, you should be able to successfully calculate the electric field both inside and outside the sphere.