Question

If the electric field is given by $\left( {5i + 4j + 9k} \right)$ , the electric flux through a surface of area $20$ unit lying in the Y-Z plane will be:
(A) $100$ unit
(B) $80$ unit
(C) $180$ unit
(D) $20$ unit

Answer 1

Hint: The flux is usually described as how much a quantity is passing through an area. Thus, the dot product of the vector with the area vector will be calculated as the vector. Here the dot product of electric fields with the area vector is to be calculated.

Complete step by step solution:
The number of electric field lines passing through an area is termed as the electric flux. If the area is larger the electric flux will be greater since more electric field lines are passing. Thus, the dot product of the two vectors giving a scalar quantity will be the electric flux. The unit of electric flux is newton meters squared per coulomb.
Given the vector of electric field as $5i + 4j + 9k$. Also given the surface lies in the X-Y plane. Therefore, the area vector will be in Z direction. Therefore, $\vec A = 20k$ .
The expression for the electric flux is given as,
$\phi = \vec E.\vec A$
Where, $\vec E$ is the vector of electric flux and $\vec A$ is the area vector.
Taking the dot product of the two vectors will give electric flux.
$ \Rightarrow \phi = \left( {5i + 4j + 9k} \right).\left( {0i + 0j + 20k} \right) \\
   \Rightarrow \left( {5 \times 0} \right) + \left( {4 \times 0} \right) + \left( {9 \times 20} \right) \\
\Rightarrow 180 $
Thus, the electric flux is $180$ units.

The answer is option C.

Note: The electric flux is directly proportional to the vector form of the electric field and the vector form of the surface area. As the electric field and the surface area is increased, then the electric flux is also increasing. As the electric field and the surface area decreases, then the electric flux is also decreasing.