Electric field in the overlap of two solid, uniformly charged spheres

• KaiserBrandon
In summary, the electric field in the overlap region is constant and has a magnitude of (r\rho)/(3\epsilon).
KaiserBrandon

Homework Statement

Two spheres, each of radius R and carrying uniform charge densities +$$\rho$$
and $$-\rho$$, respectively, are placed so that they partially overlap.
Call the vector from the positive centre to the negative centre $$\vec{d}$$. Show
that the field in the region of overlap is constant and find its value. Use
Gauss’s law to find the electric field inside a uniformly charged sphere
first.

Homework Equations

law of superposition
Gauss Law

The Attempt at a Solution

I found the field inside one sphere to be
$$(r\rho)/(3\epsilon)$$
in the radial direction. Now for the overlapping spheres, I said that the vector from the centre of the positive sphere to some point P in the interlapping area is $$\vec{r}$$. And from P to the centre of the negative sphere, I denoted $$\vec{r'}$$. so $$\vec{r'}=\vec{d}-\vec{r}$$. So in order for P to be inside the spheres, $$|\vec{r}|<R$$ and $$|\vec{d}-\vec{r}|<R$$. So using the law of superposition, inside the overlap, the electric is
$$E = (|\vec{r}|-|\vec{d}-\vec{r}|)\rho/3\epsilon$$
in the radial direction, with the boundaries in effect. Now I am stumped here, as I'm unsure how to reduce this to a constant. Any suggestions?

the electric field is a vector so why on Earth are you reducing r and d-r to scalars?

yep, realized my mistake while sitting in my thermodynamics class this morning. It's funny how I usually figure stuff out while I'm not actually trying to do the question.

k, so I changed the E function to Cartesian coordinates. So in the overlap I got:

$$\vec{E}=\frac{\rho*d}{3*\epsilon}*\hat{i}$$

where d is the magnitude of $$\vec{d}$$

And this is under the condition that $$\vec{d}$$ runs along the x axis.

sometimes you just need to sleep on it and get a fresh perpective on it in the morning

1. What is the electric field in the overlap of two uniformly charged spheres?

The electric field in the overlap of two uniformly charged spheres is a combination of the electric fields from both spheres. The total electric field at any point is the vector sum of the individual electric fields at that point.

2. How is the electric field affected by the distance between the two spheres?

The strength of the electric field in the overlap of two uniformly charged spheres is inversely proportional to the distance between the spheres. As the distance between the spheres increases, the electric field decreases.

3. Is the electric field affected by the charge of the spheres?

Yes, the electric field in the overlap of two uniformly charged spheres is directly proportional to the charge of each sphere. As the charge of the spheres increases, the electric field also increases.

4. Can the electric field be negative in the overlap of two charged spheres?

Yes, the electric field can be negative in the overlap of two charged spheres if the charges on the spheres are opposite in sign. In this case, the electric field will point in the direction of the smaller charge.

5. How can the electric field in the overlap of two charged spheres be calculated?

The electric field can be calculated using Coulomb's law, which states that the electric field at a point is equal to the product of the charges on the spheres divided by the distance squared between them. This can be written as E = k*q1*q2/r^2, where k is the Coulomb's constant, q1 and q2 are the charges on the spheres, and r is the distance between them.

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