What Is the Electric Field in the Overlapping Region of Two Charged Spheres?

[th5418]

Homework Statement

Two spheres, each radius R and carrying uniform charge densities +rho and -rho are placed so that they partially overlap. Call the vector from the positive center to the negative center dhat. Show that the field in the region of overlap is constant and find its value.

Homework Equations

Gauss' law.

The Attempt at a Solution

So I did Gauss' law for one sphere to find e-field. What I got was

E=(rho*r)/(3*episolon)

So the e-field from the positive sphere is E=(rho*r)/(3*episolon)
e-field from negative is the opposite of course.

principle of super position, don't they add up to zero?

 

[Doc Al]

So the e-field from the positive sphere is E=(rho*r)/(3*episolon)
e-field from negative is the opposite of course.

What direction does the field from each sphere point? In the area of overlap, do the fields point in opposite directions?

 

[tomcue]

Yes, according to the principle of superposition, the electric fields from the two spheres will add up to zero in regions where they do not overlap. However, in the region of overlap, the electric fields will not add up to zero and will instead create a constant electric field. This is because in the region of overlap, the electric fields from the two spheres are not acting in opposite directions and thus do not cancel out. Instead, they will combine to create a constant electric field with a magnitude equal to the sum of the individual electric fields from each sphere. Therefore, the value of the electric field in the region of overlap can be found by simply adding the electric fields from the two spheres.