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In Griffiths' Electrodynamics, it is shown that if a neutral atom is placed in an electric field, it will become polarized with a dipole moment given by $\mathbf{p}=\alpha \mathbf{E}$, where $\alpha$ is the atomic polarizability.

Then, it states that if a polar molecule is placed in an electric field, it will experience a torque: $\mathbf{N}=\mathbf{p} \times \mathbf{E}$.

My question is, why doesn't the induced dipole also experience a torque in an electric field? My reasoning is because the induced dipoles are automatically aligned in the field direction, but I am not sure if this is the right explanation.

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Your reasoning is correct. If the polarization points in the same direction as the electric field, the vector product in the formula for the torque vanishes.

In general however, one might be dealing with an anisotropic medium in which the polarization is not aligned with the field. The polarizability is then not a scalar but a matrix. The induced dipoles will then experience a torque until they have aligned with the field.

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