Electrostatic potential energy of a dipole in a uniform electric field

When a dipole of moment p is placed in a uniform electric field E, it experiences a torque tau = pE sin(theta) tending to rotate it toward alignment with the field (section 1.4.3); rotating the dipole against this torque, from one angle to another, requires an external agent to do work, and this work is stored as electrostatic potential energy of the dipole-field system. Taking the reference orientation theta0 = 90 degrees (dipole moment perpendicular to the field) as the zero of potential energy, the work needed to rotate the dipole from 90 degrees to a general angle theta is W = integral from 90 degrees to theta of pE sin(theta') d(theta') = pE[-cos(theta) - (-cos 90 degrees)] = -pE cos(theta). This work equals the potential energy at angle theta, giving U(theta) = -pE cos(theta), or in vector form, U = -p . E (minus the dot product of the dipole moment and the field). This result shows that the potential energy is most negative (minimum, the most stable configuration) when the dipole is exactly aligned with the field (theta = 0, cos theta = 1, U = -pE), and most positive (maximum, the least stable configuration) when the dipole is exactly anti-aligned with the field (theta = 180 degrees, cos theta = -1, U = +pE) -- consist …