A surface has the area vector$\overrightarrow{\mathbf{A}}\mathbf{=}\mathbf{(}\mathbf{2}\widehat{\mathbf{i}}\mathbf{+}\mathbf{3}\widehat{\mathbf{j}}\mathbf{)}{\mathbf{\text{m}}}^{\mathbf{2}}$. What is the flux of a uniform electric field through the area if the field is (a)$\overrightarrow{\mathbf{E}}\mathbf{=}\mathbf{4}\widehat{\mathbf{i}}\mathbf{\text{N}}\mathbf{/}\mathbf{\text{C}}$and (b)$\overrightarrow{\mathbf{E}}\mathbf{=}\mathbf{4}\widehat{\mathbf{j}}\mathbf{\text{N}}\mathbf{/}\mathbf{\text{C}}$.

The given area vector of the surface,$\overrightarrow{A}=(2\widehat{i}+3\widehat{j}){\text{m}}^2$

The electric field,$\overrightarrow{E}=4\widehat{i}\text{N}/\text{C}$

The electric field,$\overrightarrow{E}=-4\widehat{j}\text{N/C}$

Electric flux is the property of an electric field that describes the number of electric field lines crossing through a given surface area.

Formula:

The electric flux through a given surface,

${\varphi }_E=\overrightarrow{E}.\overrightarrow{A}$…..(i)

The amount of electric flux through the given area if the electric field is given as

$\overrightarrow{E}=4\widehat{i}\text{N}/\text{C}$

His is calculated using the given data in equation (i) as follows:

$\begin{array}{c}{\varphi }_E=(4\widehat{i}\text{N}/\text{C})\left((2\widehat{i}+3\widehat{j}){\text{m}}^2\right)\\ =8.0\text{N}\cdot {\text{m}}^2/\text{C}\end{array}$

Hence, the value of the electric flux is $8.0\text{N}\cdot {\text{m}}^2/\text{C}$.

The amount of electric flux through the given area if the electric field is given as.

$\overrightarrow{E}=4\widehat{j}\text{N/C}$

This is calculated using the given data in equation (i) as follows:

$\begin{array}{c}{\varphi }_E=(4\widehat{j}\text{N}/\text{C}).\left((2\widehat{i}+3\widehat{j}){\text{m}}^2\right)\\ =12.0\text{N}\cdot {\text{m}}^2/\text{C}\end{array}$

Hence, the value of the electric flux is $12.0\text{N}\cdot {\text{m}}^2/\text{C}$.