Overlapping spheres of charge: finding the E field between them

• jerro
Therefore, the E field in the region of overlap is zero.In summary, using Gauss' Law and the given information about the two spheres, it can be determined that the E field in the region of overlap is zero due to the uniform positive and negative charge densities balancing each other out. This is because the distance to the center of each sphere is different, resulting in different values for the E field.
jerro

Homework Statement

You have two spheres. The first is centered at the origin and as uniform positive charge density ρ and radius R. The second is shifted up a distance d, and it has uniform negative charge density -ρ and radius R.

Find the E field in the region of overlap.

Gauss' Law

The Attempt at a Solution

I first found an expression for the E field inside a single sphere.

∫E dA = $\frac{Q}{\epsilon}$
E(4$\pi$*$r^{2}$) = $\frac{\rho*(4/3)\pi*r^{3}}{\epsilon}$
E=$\frac{\rho*r}{3\epsilon}$

Now, for extending the case to include both spheres.

I add the E field from one to the E field of the other, giving $\frac{\rho*r}{3\epsilon}$ - $\frac{\rho*r}{3\epsilon}$, which gives zero.

I'm not sure that this is correct, I'm feeling weary of r, the radius of the Gaussian surface, and whether it is the same for both spheres.

r is the distance to the center of the sphere, and that is different for those spheres.
In addition, both E and r are vectors.

1. How do overlapping spheres of charge affect the electric field between them?

The overlapping spheres of charge create a combined electric field that is the sum of the individual electric fields of each sphere.

2. How can the magnitude and direction of the electric field be calculated between overlapping spheres of charge?

The magnitude and direction of the electric field can be calculated using the superposition principle, which states that the total electric field at a point is the vector sum of the individual electric fields at that point.

3. What is the formula for calculating the electric field between overlapping spheres of charge?

The formula for calculating the electric field between overlapping spheres of charge is E = (k * Q) / r², where k is the Coulomb's constant, Q is the charge of the sphere, and r is the distance between the center of the spheres.

4. How does the distance between the overlapping spheres of charge affect the electric field between them?

The electric field between overlapping spheres of charge decreases as the distance between the spheres increases. This is because the electric field follows an inverse square law, meaning that it decreases by the square of the distance.

5. Can the electric field between overlapping spheres of charge ever be zero?

Yes, the electric field between overlapping spheres of charge can be zero if the charges on the spheres are equal and opposite in sign, and the spheres are placed at a specific distance from each other. This is known as the neutral point, where the electric fields of the two spheres cancel each other out.

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