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?

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  • $\begingroup$ From Wikipedia I see $\vec{P} = \varepsilon_0 \chi \vec{E}$ in International System units. Maybe your textbook uses cgs units... $\endgroup$ – 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 $4\pi$.

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  • $\begingroup$ I know this sir freecharly. But my question was about dimensions of X not its value. P vector has dimensions Coulomb per meter square and E vector has dimensions volt per meter. So how X could be dimensionless because dimensions of P vector are not cancelled by dimensions of E vector. $\endgroup$ – Gagan Saggu Oct 10 '16 at 16:10
  • $\begingroup$ @Gagan Sagu - The electric susceptibility πœ’ is dimensionless. You can easily recognize this fact by looking at the above equation in the SI system that defines πœ’ as: $\endgroup$ – freecharly Oct 10 '16 at 19:30
  • $\begingroup$ @Gagan Sagu - continued P = πœ€0·πœ’·**E**, where πœ€0 is the free space permittivity. As you said the the dimension of P is Cm^-2. E has the dimension V·m^-1, and the free space permittivity πœ€0 has the dimension C·V^-1·m^-1. It follows from the above equation of definition that πœ€0 E has the same dimension as the polarization vector P, namely Cm^-2. Thus the susceptibility πœ’ is dimensionless. $\endgroup$ – freecharly Oct 10 '16 at 19:47
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    $\begingroup$ P and E have the same dimensions in cgs. $\endgroup$ – Rob Jeffries Oct 10 '17 at 23:08

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