1. Write the given relation

   The electric flux is given as $\phi = \alpha \sigma + \beta \lambda$ where:

   • $\sigma$ = surface charge density
   • $\lambda$ = linear charge density

2. Use dimensions of each quantity

   Since both terms on the right-hand side are added, each term must have the same dimensions as electric flux $\phi$.

   So, $[\alpha \sigma] = [\beta \lambda] = [\phi]$

3. Find dimensions of $\alpha$ and $\beta$

   From $[\alpha] = \frac{[\phi]}{[\sigma]}$ and $[\beta] = \frac{[\phi]}{[\lambda]}$

   Therefore, $\left[\frac{\alpha}{\beta}\right] = \frac{[\phi]/[\sigma]}{[\phi]/[\lambda]} = \frac{[\lambda]}{[\sigma]}$

4. Substitute dimensions of charge densities

   • Linear charge density: $[\lambda] = \frac{Q}{L}$
   • Surface charge density: $[\sigma] = \frac{Q}{L^2}$

   Hence, $\left[\frac{\alpha}{\beta}\right] = \frac{Q/L}{Q/L^2} = L$

5. Interpret the result

   Since $L$ represents length, $\dfrac{\alpha}{\beta}$ has the dimension of displacement.

6. Check options

   • A: displacement $\rightarrow$ correct
   • B: charge $\rightarrow$ incorrect
   • C: electric field $\rightarrow$ incorrect
   • D: area $\rightarrow$ incorrect

Therefore, the correct answer is A.