Question

A cone of radius $R$ and height h is located in a uniform electric field $\overrightarrow{\text{E}}$ parallel to its base. The electric flux entering the cone is:
A. $4EhR$
B. $\dfrac{1}{2}EhR$
C. $2EhR$
D. $EhR$

Answer 1

Hint: The electric flux can be defined as the vector dot product of electric field and area vector. Determine the area vector of the surface through which the electric field passes and carefully determine the angle between electric field and area vector.

Formula Used: $\phi =EA\cos \theta $

Complete answer:
Electric flux is defined as the number of electric field lines passing through a surface. It is defined as the dot product of electric field vector and area vector. The electric flux can be mathematically written as,
$\phi =EA\cos \theta $
Here, $\phi $ is electric flux, $E$ is magnitude of electric field, $A$ is area through which electric field lines pass and $\theta $ is the angle between electric field and area vector.

Now, the question mentions that the cone is placed in a uniform electric field parallel to the base of the cone. This means that the electric field passes through the curved surface. Now if we imagine the cone in two dimensions, we would obtain a triangle with its base length as twice the radius of base of cone and height as same as that of cone. The area of the triangle would be,
$\begin{align}
 & A=\dfrac{1}{2}\times \left( 2R \right)\times h \\
 & =Rh
\end{align}$
Now, the area vector for the obtained triangle would be in the same direction as that of the electric field. Thus, $\theta =0{}^\circ $.
Now, let us determine the magnitude of electric flux for the given situation.
$\begin{align}
 & \phi =EA\cos \theta \\
 & =E\cdot \left( Rh \right)\cdot \cos 0 \\
 & =EhR
\end{align}$

Thus, the correct option is (D).

Note:
Carefully determine the surface through which the electric field lines are passing. Take care while determining the angle between the electric field vector and area vector. If we imagine a cone in two dimensions, the base length would be equal to the diameter of the base of the cone.