The bound charge density and free charge density are 12 and 6 units respectively. Calculate the susceptibility.

To calculate the susceptibility ((chi)), we first need to understand the relationship between bound charge density ((rho_b)), free charge density ((rho_f)), electric displacement field ((vec{D})), electric field ((vec{E})), permittivity of free space ((varepsilon_0)), and the susceptibility itself. The total charge density ((rho)) is the sum of the bound charge density and the free charge density:

[

rho = rho_b + rho_f

]

However, this relationship doesn’t directly give us the susceptibility. Susceptibility is more directly related to the polarization ((vec{P})) of the material, which in turn affects the bound charge density, and the electric displacement field ((vec{D})) which is related to the free charge density:

[

vec{D} = varepsilon_0 vec{E} + vec{P}

]

And the polarization ((vec{P})) can also be defined in terms of susceptibility ((chi)) and the electric field ((vec{E})):

[

vec{P} = chi varepsilon_0 vec{E}

]

Given just the bound and free charge densities, without information on the electric field or the specific medium (other than the charge densities), we’re missing a direct way to calculate susceptibility ((chi)).