Question

There are three concentric thin spheres of radius $a,b,c(a>b>c)$. The total surface charge densities on their surfaces are $\sigma ,-\sigma$ and $\sigma$ respectively. The magnitude of electric field at $r$ (distance from center) such that $a>r>b$ is
(correct answer + 1, wrong answer - 0.25)

• A. $0$
• B. $\frac{\sigma }{{\in }_0{r}^2}(b^2-c^2)$
• C. $\frac{\sigma }{{\in }_0{r}^2}(a^2+b^2)$
• D. none of these

Answer 1

The correct option is B $\frac{\sigma }{{\in }_0{r}^2}(b^2-c^2)$

Electric field at a distance $r(a>r>b)$ will be due to charges enclosed in $r$ only, & since, a sphere acts as a point charge for points outside its surface,
$\therefore E=\frac{k{Q}_c}{r^2}+\frac{k{Q}_b}{r^2}=\frac{k}{r^2}(\sigma \times 4\pi c^2+(-\sigma \left)4\pi b^2\right)$

Now we can write
$=\frac{\sigma }{{\in }_0{r}^2}(c^2-b^2)=-\frac{\sigma }{{\in }_0{r}^2}(b^2-c^2)$