Question

A spherical surface has a radius of 10 cm. It has an electric charge equivalent to 3C at its center. What is the electric flux 20 cm from the center of the sphere?

Answer 1

Since the sphere of radius r=10cm=0.1m has no charge, the electric field determines only the charge located in the center (q=3C) of the sphere. The electric flux through a sphere of radius R=20cm=0.2m will be determined by the charge in the center of the sphere and is equal (according to Gauss's law):

$$\phi=\oint_S \vec E \cdot \vec n \space dA=\oint_S E_n \space dA$$

Taking into account Coulomb's law for a point charge:

$$E_n =\frac 1 {4\pi \epsilon_0} \frac q {R^2}$$

The area of a spherical surface of radius R is equal to $4\pi R^2$. So the flux is:

$$\phi=\frac 1 {4\pi \epsilon_0} \frac q {R^2} 4\pi R^2=\frac q {\epsilon_0}=\frac {3C}{8.85\cdot 10^{-12}C^2 /N \cdot m^2} \approxeq 339 \cdot 10^9 \frac{N\cdot m^2} C$$

Answer: : the electric flux is $339 \cdot 10^9 \frac {N \cdot m^2} C$