Dipole in a Uniform External Field

The figure shows a permanent electric dipole placed inside a uniform external electric field $\vec{E}$. The field is vertical, with a single arrow labelled $\vec{E}$ pointing straight upward at the left side of the diagram.

The dipole consists of two charges: a negative charge $-q$ at the lower-left and a positive charge $+q$ at the upper-right. A dashed line connects the two charges, representing the dipole axis. The dipole moment vector $\vec{p}$ points along this axis, from the negative charge toward the positive charge. The separation between the charges is labelled $a$, so the magnitude of the dipole moment is $p = qa$.

A dotted vertical reference line passes through the centre of the dipole. The dipole axis makes an angle $\theta$ with this vertical line (which is parallel to $\vec{E}$). Thus, $\theta$ is the angle between $\vec{p}$ and $\vec{E}$.

Forces and torque:

• On $+q$: force $q\vec{E}$ acting upward (parallel to $\vec{E}$).
• On $-q$: force $-q\vec{E}$ acting downward (antiparallel to $\vec{E}$).

These two forces are equal in magnitude but opposite in direction, so the net force on the dipole is zero. However, because the forces act at different points, they form a couple that produces a torque. A curved arrow (torque arc) at the centre of the dipole indicates the direction of rotation — it tends to turn $\vec{p}$ toward the direction of $\vec{E}$.

Key formula derived from this figure:

The magnitude of the torque is:

The vector form, using the cross product, is:

Here:

• $\vec{\tau}$ = torque on the dipole (direction: perpendicular to the plane of the paper, coming out of it).
• $\vec{p}$ = electric dipole moment (magnitude $p = qa$, direction from $-q$ to $+q$).
• $\vec{E}$ = uniform external electric field.
• $\theta$ = angle between $\vec{p}$ and $\vec{E}$.

Physical idea: …

What the Figure Shows

The figure has two stacked panels, (a) and (b), each depicting a dipole (charges $+q$ and $-q$ separated by distance $2a$) placed in a non-uniform electric field $\mathbf{E}$. The field is represented by plain labelled arrows that increase in length toward the right, indicating that the field strength increases in that direction. Each panel also shows a labelled arrow for the direction of increasing field and another for the direction of net force.

• Panel (a): The dipole moment $\mathbf{p}$ is parallel to $\mathbf{E}$. The negative charge $-q$ is on the left, and the positive charge $+q$ is on the right. The force on $+q$ (rightward, in the stronger part of the field) is larger than the force on $-q$ (leftward, in the weaker part). The net force therefore points rightward — toward the increasing field. The forces are drawn above the dipole.

• Panel (b): The dipole moment $\mathbf{p}$ is antiparallel to $\mathbf{E}$. Now $+q$ is on the left and $-q$ on the right. The force on $-q$ (leftward, in the stronger field) dominates, so the net force points leftward — opposite the direction of increasing field. The forces are drawn below the dipole.

Physical Idea

In a uniform field, the net force on a dipole is zero — only a torque exists. But when the field is non-uniform, the forces on the two charges are unequal in magnitude, producing a net force. The direction of this net force depends on the orientation of $\mathbf{p}$ relative to $\mathbf{E}$:

• If $\mathbf{p}$ is parallel to $\mathbf{E}$, the dipole is pulled toward stronger field.
• If $\mathbf{p}$ is antiparallel to $\mathbf{E}$, the dipole is pushed toward weaker field.

This explains why a charged comb attracts uncharged paper: the comb’s non-uniform field induces a dipole in the paper (with $\mathbf{p}$ parallel to $\mathbf{E}$), and the net force pulls the paper toward the comb.

Key Formula

The net force on a dipole in a non-uniform field is derived from the forces on each charge. For a dipole of moment $\mathbf{p} = q \cdot 2a \,\hat{\mathbf{p}}$ (where $2a$ is the separation and $\hat{\mathbf{p}}$ is the unit vector from $-q$ to $+q$), the net force when $\mathbf{p}$ is aligned with $\mathbf{E}$ is:

Here:

• $\mathbf{p}$ = dipole moment vector (magnitude $p = q \cdot 2a$)
• $\nabla$ = gradient operator (measures how $\mathbf{E}$ changes in space)
• $\mathbf{E}$ = electric field vector …