Q.A slab of material of dielectric constant k has the same area A as the plates of a parallel plate capacitor and has thickness (3/4)d, where d is the separation of the plates. The change in capacitance when the slab is inserted between the plates is (A), (B), (C), (D) -- four algebraic options in ε0A/d and k, printed in the source but not reliably recoverable letter-by-letter from the extracted text.

[!TLDR] With t=(3/4)d, C'=ε0A/d × 4k/(k+3), so the change in capacitance works out to ΔC = 3ε0A(k-1)/[d(k+3)]. [!ANSWER] C' = 4kε0A / [d(k+3)]; equivalently ΔC = 3ε0A(k-1) / [d(k+3)].

Using the general dielectric-slab formula from section 8.10.2, $C=\dfrac{\epsilon_0A}{(d-t)+t/k}$, with $t=\tfrac{3}{4}d$: $d-t=\tfrac{d}{4}$ and $t/k=\tfrac{3d}{4k}$, so the denominator is $\dfrac{d}{4}+\dfrac{3d}{4k}=\dfrac{d}{4}\left(1+\dfrac{3}{k}\right)=\dfrac{d(k+3)}{4k}$. Hence $C'=\dfrac{\epsilon_0A}{d(k+3)/4k}=\dfrac{4k\epsilon_0A}{d(k+3)}$ -- this is the new capacitance with the slab inserted. The CHANGE in capacitance the question asks for is $\Delta C=C'-C_0=\dfrac{4k\epsilon_0A}{d(k+3)}-\dfrac{\epsilon_0A}{d}=\dfrac{\epsilon_0A}{d}\left[\dfrac{4k}{k+3}-1\right]=\dfrac{\epsilon_0A}{d}\times\dfrac{4k-(k+3)}{k+3}=\dfrac{3\epsilon_0A(k-1)}{d(k+3)}$. The four lettered options as printed in the extracted source text carry fraction fragments ("3/4", "2/3", "3/2", "4/3") that the PDF-to-text extraction corrupted too badly to reconstruct a reliable letter-by-letter match against this derived formula, so the exact lettered choice is left unconfirmed here rather than guessed (per this platform's honesty rule) -- but the new capacitance $C'=4k\epsilon_0A/[d(k+3)]$ and the resulting change $\Delta C=3\epsilon_0A(k-1)/[d(k+3)]$ are both derived directly and correctly from the section 8.10.2 formula. [!ANSWER] C' = 4kε0A / [d(k+3)]; ΔC = 3ε0A(k-1) / [d(k+3)].

Substitute t=(3/4)d into the general partial-dielectric capacitance formula C=ε0A/[(d-t)+t/k], simplify, then subtract C0=ε0A/d.

A frequent error is to treat the slab as filling the WHOLE gap (using the simpler C=kC0 special case) instead of only three-quarters of it -- the fraction (3/4)d must go into the full (d-t)+t/k denominator, not be applied as a simple multiplier on k.