The radii of two insulated metal spheres are 5 cm and 10 cm. They are charged up to potential of 10 esu and 15 esu, respectively. If the two shperes are connected with one another, what will be the loss of energy?

Explanation

Radius of the first sphere = 5cm
Capacitance, `C_1 =5` statF and potential, `V_1 = 10` statV
Charge, `Q_1 =C_1V_1 = 5xx10 = 50` statC
Radius of the second sphere = 10 cm
Capacitance, `C_2 = 10` statF and potential, `V_2 =15` statV
Charge, `Q_2 = C_2 V_2 = 10 xx15 = 150` statC
Total charge of the two spheres,
`Q= Q_1+Q_2= 50+150= 200` statC
Equivalent capacitance of the combination of two spheres,
`C = C_1+C_2= 5+10= 15` statF
If V be the common potential of the two spheres after connection, then
`V = (" total charge of the two spheres ")/(" tatal capacitance of the two spheres ") = (200)/(15) (40)/(3)` statV
Total energy of the two spheres before connection,
`E_1 = (1)/(2)C_1V_1^2+(1)/(2)C_2V_2^2`
`= (1)/(2)[5xx(10)^2+10xx(15)^2] = 1375 `erg.
Total energy of the two spheres after connection,
`E_2 = (1)/(2)CV^2 = (1)/(2)xx 15 xx((40)/(3))^2 = 1333.33` erg
Loss of energy due to connection `= 1375 - 1333.33= 41.67` erg.