Two conducting spheres of radii `r_(1)` and `r_(2)` having charges `Q_(1)` and `Q_(2)` respectively are connected to each other. There is

Options

A. No change in the energy of the system
B. An increase in the energy of the system
C. Always a decreases in the energy of the system
D. A decrease in the energy of the system unless `Q_(1)R_(2)=Q_(2)R_(1)`.

Answer

D

Explanation

The potentials of the spheres are
`V_(1)=(1)/(4piepsi_(0))(Q_(1))/(R_(1)) and V_(2)=(1)/(4piepsi_(0))(Q_(2))/(R_(2))`
If `V_(1)=V_(2)` then `(Q_(1))/(R_(1))=(Q_(2))/(R_(2)) or Q_(1)R_(2)=Q_(2)R_(2)`
and there will be no flow of current betwen them and there is no decrease of energy.
but if `(Q_(1))/(R_(1)) gt (Q_(2))/(R_(2)) or (Q_(1))/(R_(1)) lt (Q_(2))/(R_(2))`
then in both cases the current will flow and there will be a decrease in the energy and it will appear at heat. thus energy decreases unless `Q_(1)R_(2)=Q_(2)R_(1)`.