The relationship between Bulk density $$(\gamma)$$, Dry density$$(\gamma_d)$$ and water content$$(\omega)$$ for soil is:

Moisture content of soil is defined as the ratio of mass of water to the mass of solids present in the soil sample. It is represented by $$\omega$$.

$$\omega$$ = $$\frac{Mass of water}{Mass of solids} = \frac{M_{w}}{M_{s}}$$

Dry density of soil is defined as the ratio of Mass of solids to the total volume of the soil. It is represented by $$\gamma_d$$

$$\gamma_d$$ = $$ \frac{Mass of solids}{Total volume of soil} = \frac{M_{s}}{V}$$

To obtain relationship between the moisture content and dry density, multiply the numerator and denominator of expression of dry density with “M” which is mass of soil sample.

$$\gamma_d$$ =  $$\frac{M_{s}}{V} \times \frac{M}{M} = \frac{M}{V} \times \frac{M_{s}}{M}$$

Mass of the soil (M) is nothing but the addition of Mass of solids (Ms) and Mass of water (Mw).

$$\gamma_d$$ = $$\frac{\frac{M}{V}}{\frac{M_{s} + M_{w}}{M_{s}}}$$

Ratio of mass to the volume is bulk density of soil which is denoted as

Therefore , $$\gamma = \gamma_d(1 + \omega)$$.

So, the answer would be option a)$$\gamma = \gamma_d(1 + \omega)$$