# Problem with the dimensions of electric susceptibility

A book says that when a linear isotropic dielectric is placed in an outer electric field $\mathbf{E}$, then the polarization per unit volume of the dielectric is $\mathbf{P}=\chi\mathbf{E}$, where $\mathbf{P}$ and $\mathbf{E}$ are vector quantities, and $\mathbf{P}$ is the polarization per unit volume of the dielectric. Then the book says that $\chi$, which is electric susceptibility of the dielectric medium, is a dimensionless quantity. How can this be?

• From Wikipedia I see $\vec{P} = \varepsilon_0 \chi \vec{E}$ in International System units. Maybe your textbook uses cgs units... – Matteo Oct 10 '17 at 22:43

In SI units the electric susceptibility $\chi$ is defined by the equation $\mathbf{P} = \epsilon_0 \chi \mathbf{E}$ which relates the polarization per unit volume $\mathbf{P}$ to the electric field $\mathbf{E}$ and $\epsilon_0$ is the free space permittivity.The electric susceptibility $\chi$ is a dimensionless quantity related to the relative permittivity $\epsilon$ of a material by $\chi = \epsilon-1$. In Gaussian units you have the susceptibility defined by $\mathbf{P} = \chi\mathbf{E}$. The electric susceptibility is dimensionless in both unit systems. Note: Although the electric susceptibilities are dimensionless in both unit systems they do not have the same numerical value. The value of the electric susceptibility in SI units differs from the electric susceptibility in Gaussian units by a factor of $4\pi$.