The figure(Figure 1) shows a solid metal sphere at the center of a hollow metal sphere. Assume that E=1.2×104N/C.
What is the total charge on the exterior of the inner sphere?
The total charge on the exterior of the inner sphere is 0, since the inner sphere is a solid metal sphere and therefore has no charge.
To find the total charge on the exterior of the inner sphere, we need to consider the electric field inside and outside the inner sphere.
First, let's understand that in a conductor, such as a metal sphere, the charges distribute themselves evenly on the outer surface. This is due to the fact that electric charges repel each other and try to get as far away as possible.
In this case, since the solid metal sphere is at the center of a hollow metal sphere, the electric field inside the hollow sphere is zero. The charges on the inner surface of the outer sphere rearrange themselves in such a way that the electric field inside the hollow sphere cancels out.
Considering this, the total charge on the exterior of the inner sphere is equal to the charge on the inner surface of the outer sphere.
To find this charge, we need to use Gauss's law. Gauss's law states that the electric flux through a closed surface is proportional to the total charge enclosed by that surface.
We will consider a spherical Gaussian surface just outside the outer sphere. Since the electric field inside the hollow sphere is zero, the flux through this Gaussian surface will also be zero.
The electric flux through the Gaussian surface is given by the formula:
Flux = E * A,
where E is the electric field and A is the surface area of the Gaussian surface.
Since the electric field outside the outer sphere is the same as the electric field inside the inner solid sphere (E = 1.2×10^4 N/C), we can write:
Flux = (1.2×10^4 N/C) * A.
The electric flux through the Gaussian surface is also related to the charge enclosed by the Gaussian surface by Gauss's law. The charge enclosed by the Gaussian surface is equal to the total charge on the exterior of the inner sphere.
Therefore, we can set up the equation:
Flux = (1.2×10^4 N/C) * A = (Total charge on the exterior of the inner sphere) / ε₀,
where ε₀ is the permittivity of free space.
Solving for the total charge on the exterior of the inner sphere:
Total charge on the exterior of the inner sphere = (1.2×10^4 N/C) * A * ε₀.
Note that the surface area A of the Gaussian surface is equal to the surface area of the outer sphere.
You can find the surface area of a solid sphere using the formula:
A = 4 * π * r^2,
where r is the radius of the sphere.
So, to find the total charge on the exterior of the inner sphere, you need to know the radius of the outer sphere and the permittivity of free space (ε₀). Plug in the values into the equation and calculate the result.
To determine the total charge on the exterior of the inner sphere, we need to consider the electric field between the two spheres. The electric field between two concentric spheres can be calculated using the formula:
E = (k * Q) / r^2
where E is the electric field, k is the electrostatic constant (k = 8.99 × 10^9 N m^2/C^2), Q is the charge on the sphere, and r is the radius of the sphere.
In this case, we have the electric field (E) given as 1.2 × 10^4 N/C. We need to calculate the charge (Q) on the inner sphere.
To find the charge on the inner sphere, we can rearrange the formula above:
Q = (E * r^2) / k
To solve this equation, we need the radius of the inner sphere. If the figure doesn't provide it, you may need to provide that information in order to proceed.