Let be an electrostatic field. Use Gauss's Law to find the total charge enclosed by the closed surface consisting of the hemisphere and its circular base in the xy-plane.

Question

Let  be an electrostatic field. Use Gauss's Law to find the total charge enclosed by the closed surface consisting of the hemisphere  and its circular base in the xy-plane.

Quizplus · Accepted Answer

According to Gauss's Law, the total charge enclosed by a closed surface is equal to the flux of the electric field through that surface divided by the permittivity of free space (ε0).

Since the enclosed surface is a hemisphere with a circular base in the xy-plane, we can choose a spherical Gaussian surface which encloses the hemisphere and is centered at the origin. The electric field is radial and has the same magnitude at all points on the Gaussian surface. Therefore, the flux of the electric field through the Gaussian surface can be calculated as the product of the magnitude of the electric field and the surface area of the Gaussian sphere.

The magnitude of the electric field can be found by using Coulomb's law:
E = k*q/r^2, where k is Coulomb's constant, q is the charge enclosed by the hemisphere, and r is the distance between the charge and the center of the Gaussian sphere.

Since the charge density is constant throughout the hemisphere, we can calculate the total charge by finding the charge density multiplied by the volume of the hemisphere:
q = (3/4)*π*r^3*(ρ)

Substituting this expression for q into the expression for the electric field, we get:
E = k*(3/4)*π*r*(ρ)/r^2 
E = (3/4)*k*π*(ρ)/r

The flux of the electric field through the Gaussian surface is then:
Φ = E*A = (3/4)*k*π*(ρ)*4π*r^2 
Φ = 3*k*π*(ρ)*r^2

Dividing this by ε0, we get:
q = Φ/ε0 = 3*k*π*(ρ)*r^2/ε0

Substituting the values of k and ε0, we get:
q = (3/4)*π*(ρ)*r^2

Therefore, the total charge enclosed by the hemisphere and its circular base is proportional to the radius squared and the charge density. The correct answer is option C.

Let Be an Electrostatic Field