Question

A uniform line charge with linear density $\lambda$ lies along the y-axis. What flux crosses a spherical surface centred at he origin with $r=R$

• A. $\frac{2R\lambda }{{ϵ}_0}$
• B. $\frac{R\lambda }{{ϵ}_0}$
• C. $\frac{\lambda }{{ϵ}_0}$
• D. none of the above

Answer 1

The correct option is B $\frac{R\lambda }{{ϵ}_0}$

Line charge density $=\lambda$

radius, $r=R$

Now, $\lambda dl=d\theta$

for total charge enclosed by radius R, ${\int }_O^{R}Q={\int }_O^R\lambda dl$

$Q_{inc}=\lambda R$

Now, $\varphi =\frac{Q_{inc}}{{\epsilon }_0}$ {from Gauss law}

$=\frac{\lambda R}{{\epsilon }_0}$ (Option B is the correct option)