Question

The number electric field lines of force emerging out of a closed surface is 1000, and then the charge enclosed by the surface is:
$\begin{align}
  & (A)8.854\times {{10}^{-9}}C \\
 & (B)8.854\times {{10}^{-4}}C \\
 & (C)8.854\times {{10}^{-1}}C \\
 & (D)8.854C \\
\end{align}$

Answer 1

Hint: The number of electric field lines of force emerging out of a closed surface represents the flux and we need to find the charge enclosed. The Gauss law of electrostatics relates the amount of charge enclosed with flux, so it is ideal to use the Gauss law in this case.

Formula used:
$\phi =\dfrac{q}{{{\varepsilon }_{0}}}$
Where q is the enclosed charge

Complete step by step answer:
The Gauss law in electrostatics states that the total flux out of a closed surface is equal to the charge enclosed divided by the permittivity of free space.
$\phi =\dfrac{q}{{{\varepsilon }_{0}}}$
$q$ is the enclosed charge
$\phi $ is the electric flux

Given,
No of lines emerging out of the closed surface=1000
To find: The charge enclosed by the closed surface
According to the Gauss law of electrostatics, the electric flux and charge are related as follows:
$\phi =\dfrac{q}{{{\varepsilon }_{0}}}$
Here, electric flux is given by the number of lines emerging out of the closed surface, which is equal to 1000
Plugging the value of $\phi $ and ${{\varepsilon }_{0}}$ which is a constant, we have
$\begin{align}
  & \Rightarrow \phi =\dfrac{q}{{{\varepsilon }_{0}}} \\
 & \Rightarrow 1000=\dfrac{q}{8.8419\times {{10}^{-12}}} \\
 & \Rightarrow q=8.8419\times {{10}^{-9}}C \\
\end{align}$
The enclosed charge can thus be given as $(A)8.854\times {{10}^{-9}}C$

Additional Information:
By the definition of electric flux, it is clear that it is given by the number of field lines emerging out of a closed surface. Electric flux is a property of electric fields. It describes the influence of electric field lines over a given surface. The relation called the Gauss’s law is only valid for closed Gaussian surfaces and cannot be applied elsewhere.


Note:
Gauss law of electrostatics is one of the four Maxwell equations. In this problem, it is applied to calculate the electric charge when the number of field lines emerging out of a closed surface is given. Conversely, it can also be applied to calculate the flux or the number of field lines coming out of a closed surface when the total charge enclosed by the closed surface is given.