A sphere of capacitance 100 pF is charged to a potential of 100 V....

Initial energy of the charged sphere is given by:

$U_i = \dfrac{1}{2} C V^2$

$U_i = \dfrac{1}{2} \times 100 \times 10^{-12} \times (100)^2 = 5 \times 10^{-7}$ J

When the identical uncharged sphere is brought in contact, the charge is shared equally and the common potential becomes:

$V' = \dfrac{V}{2} = 50$ V

Final total energy of the two spheres is:

$U_f = 2 \times \left( \dfrac{1}{2} C V'^2 \right) = C V'^2$

$U_f = 100 \times 10^{-12} \times (50)^2 = 2.5 \times 10^{-7}$ J

Change in energy is the loss of energy during charge sharing:

$\Delta U = U_i - U_f = 5 \times 10^{-7} - 2.5 \times 10^{-7} = 2.5 \times 10^{-7}$ J

$\Delta U = \dfrac{5}{2} \times 10^{-7}$ J

Comparing with $\alpha \times 10^{-7}$ J, we get $\alpha = \dfrac{5}{2}$.

Answer: $\dfrac{5}{2}$