Two small spheres of masses `M_(1)` and `M_(2)` are suspended by weightless insulating threads of lengths `L_(1)` and `L_(2)`. The spheres carry charges `Q_(1)` and `Q_(2)`, respectively. The spheres are suspended such that they are in level with one another and the threads are inclined to the vertical at angle of `theta_(1)` and `theta_(2)` as shown. Which one of the following conditions is essential, if `theta_(1) = theta_(2)`?

Options

A. `m_(1)=m_(2)`
B. `|q_(1)|=|q_(2)|`
C. `l_(1)=l_(2)`
D. `(q_(1))/(m_(1))=(q_(2))/(m_(2))`

Answer

A

Explanation

Since both the small spheres are at same horizontal level, the electrostatic forces on both spheres are in horizontal direction. The FBD of left sphere is shown in figure.
`:'` The sphere is equilibrium `rArr T cos theta= mg and T sin theta=F_(E)`
`:. tan theta=(F_(E))/(mg)`

The magnitude of electrostatic force on each sphere is same irrespective of its charge `:. ` for `theta_(1)=theta_(2)` the necessary condition is `m_(1)=m_(2)`