Question

$q_1,q_2,q_3,\text{and}{q}_4$ are point charges located at points as shown in the figure and S is a spherical Gaussian surface of radius R. Which of the following is true according to the Gauss's law?

• A. ${\oint }_5\left({\overrightarrow{E}}_1+{\overrightarrow{E}}_2+{\overrightarrow{E}}_3\right).d\overrightarrow{A}=\frac{q_1+q_2+q_3}{2{\epsilon }_0}$
• B. ${\oint }_{\left({\overrightarrow{E}}_1+{\overrightarrow{E}}_2+{\overrightarrow{E}}_3\right).d\overrightarrow{A}=\frac{\left(q_1+q_2+q_3\right)}{{\epsilon }_0}}$
• C. ${\oint }_5\left({\overrightarrow{E}}_1+{\overrightarrow{E}}_2+{\overrightarrow{E}}_3\right).d\overrightarrow{A}=\frac{\left(q_1+q_2+q_3+q_4\right)}{{\epsilon }_0}$
• D. None of the above

Answer 1

The correct option is B ${\oint }_{\left({\overrightarrow{E}}_1+{\overrightarrow{E}}_2+{\overrightarrow{E}}_3\right).d\overrightarrow{A}=\frac{\left(q_1+q_2+q_3\right)}{{\epsilon }_0}}$

$\oint \overrightarrow{E}.\overrightarrow{ds}=\frac{q_{enclosed}}{{\epsilon }_0}\oint (\overrightarrow{E_1}+\overrightarrow{E_2}+\overrightarrow{E_3}).d\overrightarrow{A}=\frac{(q_1+q_2+q_3)}{{\epsilon }_0}$