Two spheres of radii $5\; cm$ and $10\;cm$ are given charges $100\;C$ and $50\;C$ respectively and then connected by a wire. Calculate the loss of energy after connection.

Given,
Radius of $1^{st}$ sphere $(r_1) = 5\;cm$
Radius of $2^{st}$ sphere $(r_2) = 10\;cm$
Charge on 1^{st} sphere $(q_1) = 100\;C$
Charge on $2^{nd}$ sphere $(q_1) = 100\;C$
Loss of energy $(\Delta E) = ?$
Capacitance of first sphere

$C_1 = 4\; \pi \; \epsilon_0 \; r_1 = 4 \pi \;*\; 10^{-12}\;*\;0.05 = 5.56 * 10^{-12}\;F$

Capacitance of second sphere

$C_2 = 4\; \pi \; \epsilon_0 \; r_2 = 4 \pi \;*\; 10^{-12}\;*\;0.1 = 1.11 \; *\; 10^{-12}\;F$

Then,

$V_1 = $ $\frac{q_1}{C_1} = \frac{100}{5.56\;*\;10^{-12}}$ $ = 1.8\;*\;10^{13}\;V$

$V_2 = $ $\frac{q_2}{C_2} = \frac{50}{1.11\;*\;10^{-11}}$ $ = 4.5\;*\;10^{12}\;V$

Total energy before connection,

$E_1 = \frac{1}{2}C_1\;V_1^2 + \frac{1}{2}C_2\;V_2^2$

$E_1= \frac{1}{2}* 5.56 * 10^{-12} * (1.8 * 10^{13})^2 + \frac{1}{2}* 1.11 * 10^{-11} * (4.5 * 10^{12})^2$ 

$= 1.01 * 10^{15}\;J$

Total energy after connection,

$E_2 = \frac{1}{2} \frac{(C_1\;V_1\; + \; C_2 \; V_2)^2}{C_1 \; + \; C_2} = \frac{1}{2} \frac{(5.56 * 10^{-12}\;*\; 1.8\;*\;10^{13}\; + \; 1.11 \; *\; 10^{-12} \; * \; 4.5\;*\;10^{12})^2}{5.56 \;*\; 10^{-12} \; + \; 1.11 \; *\; 10^{-12}}$ 

$ = 6.67\;*\;10^{14}\;J$

Then loss is energy,

$\Delta E = E_1 - E_2 = 1.01\;*\;10^{15} - 6.75\;*\;10^{14} = 3.6\;*\;10^{14}\;J$

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