Question
Question: A uniformly charged conducting sphere is having radius \[1\,m\] and surface charge density \(20\,C{m...
A uniformly charged conducting sphere is having radius 1m and surface charge density 20Cm−2 . The total flux leaving the Gaussian surface enclosing the sphere is
A. 40πε0−1
B. 80πε0−1
C. 20πε0−1
D. 60πε0−1
Solution
Here, we will use the concept of Gauss law in the case of the conducting sphere. Gauss law states that the electric flux linked with the surface will be equal to the ε01 times the charge enclosed in that surface. The charge in this case can be calculated by using the formula of surface charge density.
Formula used:
The formula of electric flux using Gauss law is given by
ϕ=ε0q
Here, ϕ is the electric flux, q is the charge in the sphere, and ε0 is the permittivity.
Also, the formula of surface charge density is given by
σ=Aq
Here, σ is the charge density of the sphere, q is the charge, and A is the area of the conducting sphere.
Complete step by step answer:
Consider a uniformly charged conducting sphere that is of the radius 1m.
The radius of the sphere, r=1m
Also, surface charge density, σ=20Cm−2
The electric flux in this case can be calculated by using Gauss law. Gauss law states that the total flux inside a charged sphere is equal to the ε01 times the charge enclosed in that surface.
The formula of the electric flux is given by
ϕ=ε0q
Surface charge density is defined as the charge in the sphere per unit area of the sphere. The formula of the surface charge density is given by
σ=Aq
⇒q=σA
Putting this value in the formula of electric flux as shown below
ϕ=ε0σA
Now, the area of the sphere is given by
A=4πr2
Now, the formula of electric flux is given by
ϕ=ε0σ(4πr2)
⇒ϕ=ε020(4π(1)2)
⇒ϕ=ε080π
∴ϕ=80πε0−1
Therefore, the total flux leaving the Gaussian surface enclosing the sphere is 80πε0−1 .
Hence, option B is the correct option.
Note: Here, in the above question ε0 is the permittivity in free space. The value of permittivity in free space is ε0=8.8×10−12m−3kg−1s4A2 . Here, we do not have this value in the above example because we want the answer in terms of ε0 .