To account for the effect of the lens thickness t , four...

For a modified lens formula to be correct, it must reduce to the standard thin lens formula when the thickness of the lens approaches zero ($t \to 0$).

The standard thin lens formula is given by:
$\dfrac{1}{v} - \dfrac{1}{u} = \dfrac{1}{f}$

Substituting $t = 0$ into the first expression gives $\dfrac{1}{v - 0} - \dfrac{1}{u + 0} = \dfrac{1}{v} - \dfrac{1}{u} = \dfrac{1}{f}$, which matches the standard formula.

Substituting $t = 0$ into the second expression gives $\dfrac{1}{v} - \dfrac{1}{u} = 0$, which is incorrect.

Substituting $t = 0$ into the third expression gives $-\dfrac{1}{u} = \dfrac{1}{f}$, which is incorrect.

Substituting $t = 0$ into the fourth expression gives $\dfrac{1}{v} - \dfrac{1}{u} = 0$, which is incorrect.

Additionally, the fourth expression is dimensionally incorrect as the right-hand side $\dfrac{t}{f}$ is dimensionless, while the left-hand side has dimensions of $[L]^{-1}$.

Therefore, only the first modification is physically consistent.