Energy of an induced electric dipole If an atom with scalar polarizability α is placed in a uniform electric field of magnitude E, is the potential energy associated with the induced dipole equal to 0.5 * α * E^2?

Introduction / Context:
In dielectrics, an applied electric field induces a dipole moment p = α * E (for linear, isotropic media at low fields). The sign and magnitude of the associated potential energy determine whether a material is attracted into or repelled from regions of strong field, which is crucial in optical trapping and dielectrophoresis.

Given Data / Assumptions:

• Linear response: p = α * E, with α > 0 as a scalar for simplicity.
• Uniform field E; no spatial gradients considered in the energy expression.
• Quasi-static regime; conduction losses neglected.

Concept / Approach:
The work done to build the induced dipole from zero field to E is obtained by integrating the incremental work dU = − p · dE = − α * E · dE. This yields the potential energy of the induced dipole:

U = − ∫ (α * E) dE = − 0.5 * α * E^2.

This negative sign indicates the system lowers its energy when the field increases, consistent with attraction toward high-field regions for materials with positive α.

Step-by-Step Solution:
Start from p = α * E (linear polarization).Incremental work on the dipole: dU = − p · dE = − α * E dE.Integrate from 0 to E: U = − 0.5 * α * E^2.Therefore, the statement “U = + 0.5 * α * E^2” is incorrect; the correct expression has a negative sign.

Verification / Alternative check:
Energy densities from macroscopic electrodynamics give u = 0.5 * E · D in dielectrics, which when decomposed into vacuum and polarization parts leads to the same −0.5 * α * E^2 for the microscopic induced dipole.

Why Other Options Are Wrong:
Frequency or crystal type does not change the sign of the potential energy for a linear induced dipole. Anisotropy changes α to a tensor but not the sign after proper tensor contraction.

Common Pitfalls:
Dropping the minus sign when integrating or confusing the stored field energy density with the dipole potential energy in an external field.

Final Answer:
False (the potential energy is -0.5 * α * E^2)