Electrostatics

Q? = CV?
U? = ½CV? ²
q? = (q? C)/ (C + C/2) = 2q? /3

q? = (q? (C/2)/ (C + C/2) = q? /3; Uf = q? ²/2C + q? ²/2 (C/2)
= (4q? ²/9x2C) + (q? ²/9C)
= (6q? ²/18C) = q? ²/3C = CV? ²/3

Energy loss in the process = U? – Uf
= ½CV? ² - CV? ²/3
= CV? ²/6

$U_{\text{initial}}=\frac{k\left(4q\right)\left(q\right)}{(d/2)}+\frac{k\left(q\right)(-q)}{(d/2)}$

$\begin{array}{rr}& \Rightarrow \mathrm{}\frac{6k{q}^2}{d}\\ & \Rightarrow \mathrm{}{U}_{\text{final}}=\frac{4\left(4q\right)\left(q\right)}{\left(\frac{3d}{2}\right)}+\frac{k\left(q\right)(-q)}{(d/2)}\\ & \Rightarrow \mathrm{}\frac{2}{3}\frac{k{q}^2}{d}\\ & \Rightarrow \mathrm{}\mathrm{\Delta }U=\left(\frac{2}{3}-6\right)\frac{k{q}^2}{d}\Rightarrow \frac{-16}{3}\frac{{\mathrm{k}\mathrm{q}}^2}{d}\end{array}$

${\sigma }_1={\sigma }_2$

$\begin{array}{rr}& \frac{{\mathrm{Q}}_1}{4\pi {\mathrm{R}}^2}=\frac{{\mathrm{Q}}_2+{\mathrm{Q}}_1}{4\pi \left(16{\mathrm{R}}^2\right)}\\ & \Rightarrow \mathrm{}{\mathrm{Q}}_1+{\mathrm{Q}}_2=16{\mathrm{Q}}_1\\ & \Rightarrow \mathrm{}{1}_1={\mathrm{Q}}_2\\ & \mathrm{V}\left(\mathrm{R}\right)-\mathrm{V}\left(4\mathrm{R}\right)\\ & =\left(\frac{{\mathrm{K}\mathrm{Q}}_1}{\mathrm{R}}+\frac{{\mathrm{K}\mathrm{Q}}_2}{4\mathrm{R}}\right)-\left(\frac{{\mathrm{K}\mathrm{Q}}_1}{4\mathrm{R}}+\frac{{\mathrm{K}\mathrm{Q}}_2}{4\mathrm{R}}\right)\\ & =\frac{3\mathrm{K}\mathrm{Q}{\mathrm{Q}}_1}{4\mathrm{R}}=\frac{3{\mathrm{Q}}_1}{16\pi \in \mathrm{R}}\end{array}$

When the two conducting spheres are connected by a wire, they will reach the same electric potential, V.
The total charge Q_total = 12µC + (-3µC) = 9µC. This total charge will redistribute.
Let the final charges be q? and q? + q? = 9µC.
The potential of a sphere is V = kq/r.
V? = V?
k q? /R? = k q? /R? ⇒ q? /R? = q? /R?
q? / (2R/3) = q? / (R/3) ⇒ q? /2 = q? ⇒ q? = 2q?
Substitute this into the charge conservation equation:
2q? + q? = 9µC ⇒ 3q? = 9µC ⇒ q? = 3µC.
Then, q? = 2 * 3µC = 6µC.
The final charges are 6µC and 3µC.