question_answer

There are three concentric thin spheres of radius a, b, c (a > b > c). The total surface charge densities on their surfaces are \[\sigma ,\]\[-\,\sigma ,\]\[\sigma \] respectively. The magnitude: of electric field at r (distance from centre) 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 :

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, \[\therefore \]      \[E=\frac{k{{Q}_{c}}}{{{r}^{2}}}+\frac{k{{Q}_{b}}}{{{r}^{2}}}=\frac{k}{{{r}^{2}}}\]             \[(\sigma \times 4\pi {{c}^{2}}+(-\,\sigma )4\pi {{b}^{2}})=\frac{\sigma }{{{\varepsilon }_{0}}{{r}^{2}}}({{c}^{2}}-{{b}^{2}})\]