There are two conducting concentric hollow spheres of outer radii and ( ). The thickness of the material of both spheres is .
The inner sphere is negatively charged with charge density . The larger sphere is positively charged with charge density .
(a) What is the electric field (magnitude and direction) inside the inner sphere?
Direction: use units vectors in spherical coordinates ( , , )
To find the electric field inside the inner sphere, we can use Gauss's Law. Gauss's Law states that the electric flux passing through any closed surface is equal to the charge enclosed divided by the permittivity of the medium.
Let's consider a Gaussian surface inside the inner sphere. Since the electric field is spherically symmetric in this case, we can use a spherical Gaussian surface. The magnitude of the electric field inside the inner sphere is the same everywhere on the Gaussian surface.
The Gaussian surface is a concentric sphere with a radius less than the inner sphere's radius. Let's call this radius "r".
Now, we need to calculate the charge enclosed within this Gaussian surface. The inner sphere, with a charge density ρ1, will have a total charge:
Q1 = ρ1 * volume of the inner sphere
The volume of the inner sphere is:
V1 = (4/3)π(r1^3)
Substituting the values, we have:
Q1 = ρ1 * (4/3)π(r1^3)
Next, let's calculate the electric flux passing through the Gaussian surface. The electric flux (Φ) is given by:
Φ = E * A
where E is the electric field and A is the surface area of the Gaussian surface.
The surface area of the Gaussian surface is:
A = 4πr^2
Substituting the values, we have:
Φ = E * 4πr^2
According to Gauss's Law, the electric flux passing through the Gaussian surface is equal to the charge enclosed divided by the permittivity of the medium (ε0):
Φ = Q1 / ε0
Substituting the values, we have:
E * 4πr^2 = (ρ1 * (4/3)π(r1^3)) / ε0
Now, we can solve for the magnitude of the electric field (E) inside the inner sphere:
E = (ρ1 * (r1^3)) / (3ε0r^2)
The direction of the electric field inside the inner sphere can be found using unit vectors in spherical coordinates (r, θ, φ). In this case, the electric field will point towards the center of the sphere or along the negative r-direction.
So, the final answer is:
The magnitude of the electric field (E) inside the inner sphere is (ρ1 * (r1^3)) / (3ε0r^2), and its direction is along the negative r-direction.