P19-140 – Electric Dipole in a Uniform Electric Field

Electric Dipole in a Uniform Electric Field

Definition: electric dipole

Two point charges $+q$ and $–q$ separated by a fixed distance $d$ constitute an electric dipole.

Electric dipoles are typically represented by their dipole moment $\overrightarrow{p}$ which is a vector that points from $-q$ to $+q$ and has magnitude $p=qd\ \ (C\cdot m)$ where $d$ represents the fixed distance between the two charges.

This vector gives you the orientation of the dipole through its direction and the strength of the dipole through its magnitude.

Motion in an electric field:

When placed in a uniform electric field, each charge of the electric dipole experiences the electric force. The charge $+q$ experiences an electric force ${\overrightarrow{F}}_{+}$ in the direction of $\overrightarrow{E}$ while the charge $-q$ experiences an electric force ${\overrightarrow{F}}_{-}$ in the direction opposite $\overrightarrow{E}$ as shown below.

Net force acting on the dipole:

The net force acting on the dipole is zero because the field is uniform and the charges are equal in magnitude. Indeed, we write

\boxed{F_{net\ x}=F_+-F_-=qE-qE=0}

Net torque acting on the dipole:

The net torque acting on the dipole, however, is not zero and will tend to align the dipole with the electric field lines in such a way that $\overrightarrow{p}$ and $\overrightarrow{E}$ are parallel.

The torque on the $+q$ charge is clockwise and equal to

{\tau }_+=-\frac{d}{2}qEsin\left(\theta \right)

The torque on the $-q$ charge is clockwise and equal to

{\tau }_-=-\frac{d}{2}qEsin\left(\theta \right)

The net torque is therefore negative (i.e., clockwise) and equal to

\boxed{{\tau }_{net}={\tau }_++{\tau }_-=-dqEsin\left(\theta \right)=-pEsin\left(\theta \right)}

It is customary to generalize the expression of the net torque acting on the dipole by writing it as the following cross product

\boxed{{\overrightarrow{\tau }}_{net}=\overrightarrow{p}\times \overrightarrow{E}}

which has magnitude

\left|{\tau }_{net}\right|=pE{\mathrm{sin} \left(\theta \right)\ }

If we neglect any energy loss in this motion, the dipole will oscillate back and forth forever. In practice, of course, the energy loss dampens the oscillations and causes the dipole to eventually align its dipole moment with the applied electric field.

Electric potential enegy of the dipole:

The electric potential energy of the dipole can be expressed as the following dot product

\boxed{U_d=-\overrightarrow{p}\cdot \overrightarrow{E}=-qdE{\mathrm{co}\mathrm{s} \left(\theta \right)\ }}

The electric potential energy of the dipole is minimized when the dipole moment $\overrightarrow{p}$ points in the same direction as the electric field $\overrightarrow{E}$ (stable equilibrium) whereas it is maximized when the dipole moment $\overrightarrow{p}$ points opposite the direction the electric field $\overrightarrow{E}$ (unstable equilibrium).

We will define electric potential energy in a later chapter and for now we accept the above as a definition of the electric potential energy of a dipole.