question_answer

There are three concentric thin spheres of radius \[a,\text{ }b,\text{ }c\,\text{(}a>b>c).\] The total surface charge densities on their surfaces are \[\sigma ,\]\[-\sigma \] and a respectively. The magnitude of electric field at r (distance from center) such that \[a>r>b\]is -

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

Correct Answer: B

Solution :

[B]

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,

\[\therefore \,\,E=\frac{k{{Q}_{c}}}{{{r}^{2}}}+\frac{k{{Q}_{b}}}{{{r}^{2}}}=\frac{k}{{{r}^{2}}}\,\,(\sigma \times 4\pi {{c}^{2}}+(-\,\sigma )4\pi {{b}^{2}})\]

\[=\frac{\sigma }{{{\varepsilon }_{0}}{{r}^{2}}}({{c}^{2}}-{{b}^{2}})\]