Question

An infinitely long time charge having a uniform charge per unit length $\lambda$ lies at a distance d from a point O as shown in Fig. Determine the total electric flux through the surface of sphere of radius R centered at O resulting from this line charge. Consider both cases where R<d and R>d.

• A. $\frac{2\lambda \sqrt{R^2-d^2}}{{\epsilon }_0}$
• B. $\frac{7\lambda \sqrt{R^2-d^3}}{{\epsilon }_0}$
• C. $\frac{7\lambda \sqrt{R^2-d^2}}{2{\epsilon }_0}$
• D. $\frac{\lambda \sqrt{R^2-d^2}}{5{\epsilon }_0}$

Answer 1

The correct option is A $\frac{2\lambda \sqrt{R^2-d^2}}{{\epsilon }_0}$

For R < d, the flux will be zero. It is because there is no charge line inside the sphere. For R < d, the length inside the sphere is
$2\left(\sqrt{R^2-d^2}\right)$.
So the charge inside is
$2\lambda \sqrt{R^2-d^2}$
Thus $\varphi =\frac{2\lambda \sqrt{R^2-d^2}}{{\epsilon }_0}$