The electric field at any points on the sphere of radius 0.75 m is equal to 8.9 x 102 N/C and points radially to the center of the sphere. What is the net charge at the center of the sphere ?
To find the net charge at the center of the sphere, we need to use Gauss's Law. Gauss's Law states that the electric flux through a closed surface is proportional to the net charge enclosed by the surface.
The electric field at any point on the surface of the sphere is given as 8.9 x 10^2 N/C and points radially to the center of the sphere. Since the electric field is radially symmetric, we can choose a spherical Gaussian surface that encloses the entire sphere.
The electric flux through a closed surface is given by the formula:
Φ = E * A
where Φ is the electric flux, E is the electric field, and A is the surface area.
The surface area of a sphere is given by the formula:
A = 4πr^2
where r is the radius of the sphere.
Substituting the values into the formula:
Φ = (8.9 x 10^2 N/C) * (4π(0.75 m)^2)
Simplifying the equation:
Φ = (8.9 x 10^2 N/C) * (4π(0.5625 m^2))
Φ = (8.9 x 10^2 N/C) * (2.25π m^2)
Φ = 19.9 x 10^2 N m^2/C
The electric flux through the entire surface of the sphere is 19.9 x 10^2 N m^2/C.
According to Gauss's Law, the electric flux through a closed surface is also equal to the net charge enclosed by the surface divided by the electric constant ε₀.
Φ = Q / ε₀
Solving for Q:
Q = Φ * ε₀
where ε₀ is the electric constant, approximately equal to 8.85 x 10^-12 C^2/(N m^2).
Plugging in the values:
Q = (19.9 x 10^2 N m^2/C) * (8.85 x 10^-12 C^2/(N m^2))
Simplifying the equation:
Q = (19.9 x 10^2) * (8.85 x 10^-12 C)
Q = 175.935 x 10^-10 C
Therefore, the net charge at the center of the sphere is approximately 175.935 x 10^-10 C.
To find the net charge at the center of the sphere, we need to use Gauss's Law. Gauss's Law states that the electric flux through a closed surface is equal to the net charge enclosed by that surface divided by the permittivity constant of the medium.
The electric flux through a closed surface is given by the formula:
Φ = E * A
Where:
Φ is the electric flux
E is the electric field
A is the area of the surface
Since the electric field is pointing radially towards the center of the sphere and the sphere is a closed surface, the electric flux through the sphere is the same at every point. Therefore, we can calculate the electric flux at any point on the sphere's surface.
Using the formula for the electric flux, we have:
Φ = E * A
where:
E = 8.9 x 10^2 N/C (given)
A = 4πr^2 (where r is the radius of the sphere)
Substituting the values, we have:
Φ = (8.9 x 10^2 N/C) * (4π(0.75)^2)
Φ = 8.9 x 10^2 N/C * 4π * 0.75^2
Φ = 8.9 x 10^2 N/C * 4 * 3.14159 * 0.5625
Φ = 3520.68 Nm^2/C
According to Gauss's Law, the electric flux through the surface of a closed object is equal to the net charge enclosed divided by the permittivity constant of the medium. In this case, since the electric flux is the same at every point on the closed surface, we can calculate the net charge enclosed by the sphere.
Φ = Q / ε0
Where:
Q is the net charge enclosed
ε0 is the permittivity constant of free space
Rearranging the equation, we have:
Q = Φ * ε0
Substituting the values, we have:
Q = (3520.68 Nm^2/C) * (8.854 x 10^-12 C^2/Nm^2)
Q = 3.12 x 10^-8 C
Therefore, the net charge at the center of the sphere is approximately 3.12 x 10^-8 C.