The three slabs are placed side-by-side (parallel combination), each occupying one-third of the area $A/3$ and the full separation $d$.

Total capacitance:

$C = \frac{\varepsilon_0 (A/3)}{d}(K_1 + K_2 + K_3) = \frac{\varepsilon_0 A}{3d}(10+12+14) = \frac{36\varepsilon_0 A}{3d} = \frac{12\varepsilon_0 A}{d}$

For a single dielectric $K$:

$C = \frac{K\varepsilon_0 A}{d}$

$\Rightarrow K = 12$

The equivalent dielectric constant is simply the arithmetic mean: $(10+12+14)/3 = 12$.

The four dielectrics fill four quadrants. Each half of the plate area (left and right halves, each of width $d/2$) acts as two capacitors in series (top and bottom dielectrics), and these two series combinations are in parallel.

More directly using the standard result for this arrangement:

$K_{eff} = \frac{(K_1+K_2)(K_3+K_4)}{K_1+K_2+K_3+K_4}$

This comes from treating left column ($K_1, K_3$ in series) in parallel with right column ($K_2, K_4$ in series), where each column occupies half the area and full thickness $d$, but each dielectric occupies half the thickness $d/2$.

Work done = $Q \times V_{origin}$, where $V_{origin}$ is the potential at origin due to the four charges.

Distances from origin:

• $(0,2)$: $r_1 = 2$
• $(0,-2)$: $r_2 = 2$
• $(4,2)$: $r_3 = \sqrt{16+4} = \sqrt{20} = 2\sqrt{5}$
• $(4,-2)$: $r_4 = \sqrt{20} = 2\sqrt{5}$

$V = \frac{Q}{4\pi\varepsilon_0}\left(\frac{1}{2}+\frac{1}{2}+\frac{1}{2\sqrt{5}}+\frac{1}{2\sqrt{5}}\right) = \frac{Q}{4\pi\varepsilon_0}\left(1 + \frac{1}{\sqrt{5}}\right)$

Work $= QV = \dfrac{Q^2}{4\pi\varepsilon_0}\left(1+\dfrac{1}{\sqrt{5}}\right)$

Since the battery is disconnected, charge $Q$ is constant:

$Q = C_0 V_0 = 12 \times 10^{-12} \times 10 = 120\ \text{pC}$

Initial energy: $U_i = \frac{Q^2}{2C_0} = \frac{(120\times10^{-12})^2}{2\times12\times10^{-12}} = 600\ \text{pJ}$

New capacitance: $C = KC_0 = 6.5 \times 12 = 78\ \text{pF}$

Final energy: $U_f = \frac{Q^2}{2C} = \frac{(120)^2 \times 10^{-24}}{2\times78\times10^{-12}} \approx 92.3\ \text{pJ}$

Work done by capacitor on slab $= U_i - U_f = 600 - 92.3 \approx 508\ \text{pJ}$