A sphere of capacitance 100 pF is charged to a potential of 100 V. Another identical uncharged metal sphere is brought in contact with the charged sphere, then the change in the total energy stored on these spheres, when they touch is $$\alpha \times 10^{-7}$$ J. The value of $$\alpha$$ is __________. (combined capacitance of spheres is 200 pF)

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

$$U_i = \frac{1}{2} \times (100 \times 10^{-12}) \times (100)^2 = \frac{1}{2} \times 10^{-10} \times 10^4 = 5 \times 10^{-7}\text{ J}$$

$$V_c = \frac{\text{Total Charge}}{\text{Total Capacitance}} = \frac{C \cdot V + 0}{C + C} = \frac{V}{2}$$

$$V_c = \frac{100}{2} = 50\text{ V}$$

$$U_f = \frac{1}{2} C_{\text{combined}} V_c^2$$

$$U_f = \frac{1}{2} \times (200 \times 10^{-12}) \times (50)^2 = 100 \times 10^{-12} \times 2500 = 2.5 \times 10^{-7}\text{ J}$$

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

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