Potential Due to a Dipole

Electric Potential Due to a Dipole

A dipole consists of two equal and opposite charges $+q$ and $-q$, separated by a small distance $d$. The electric potential due to a dipole at any point in space depends on the position of the point relative to the dipole and the orientation of the dipole.

Key Concepts:

• The dipole moment $\vec{p}$ is defined as:

  $$\vec{p} = q \cdot \vec{d}$$

  where:

  • $q$ is the magnitude of each charge,
  • $\vec{d}$ is the vector separating the two charges, pointing from the negative charge to the positive charge.

• The electric potential $V$ is a scalar quantity that describes the work done per unit charge to move a test charge from infinity to a point in the electric field. The potential due to a dipole has both magnitude and direction, depending on the location in space.

1. General Expression for Electric Potential Due to a Dipole

The electric potential $V(\vec{r})$ at a point $\vec{r}$ due to a dipole is derived from the contributions of both charges. For simplicity, we consider the dipole to be located at the origin and aligned along the $x$-axis (though the result is general and can be applied for any orientation).

The electric potential at a point $\vec{r}$ in space due to a dipole is given by the following formula:

where:

• $\vec{p} = q \cdot \vec{d}$ is the dipole moment,
• $\hat{r}$ is the unit vector in the direction from the dipole to the point $\vec{r}$,
• $r$ is the distance from the dipole to the point where the potential is being calculated.

In simpler terms, the potential at a point due to a dipole is the scalar product of the dipole moment and the unit vector $\hat{r}$ divided by $r^2$, with the constant factor $\frac{1}{4 \pi \epsilon_0}$.

2. Potential Due to a Dipole at a General Point

To calculate the potential more explicitly, we consider the position of the dipole and the point where the potential is being evaluated.

a. Potential on the Axial Line (Along the Dipole Axis)

Consider a point on the axial line of the dipole, which is the straight line that passes through the dipole and extends along the direction of the dipole moment (the $x$-axis in our case).

• For a point located at a distance $r$ along the axial line from the dipole (with $r$ much larger than the separation distance $d$, so the dipole is treated as a point dipole), the potential is:

where:

• $p = qd$ is the dipole moment,
• $r$ is the distance from the dipole along the axial line.

b. Potential on the Equatorial Line (Perpendicular to the Dipole Axis)

Consider a point on the equatorial line, which is the perpendicular bisector of the dipole (the line that is perpendicular to the dipole axis and passes through the center of the dipole).

• For a point at a distance $r$ from the dipole along the equatorial line, the potential is:

Here:

• The potential is negative, because the two charges on the dipole are opposite in sign, and the potential at points along the equator is due to the cancellation of the two charges' contributions.

3. Potential at Points Far from the Dipole (Dipole Approximation)

If the point is located at a large distance $r$ from the dipole (i.e., $r \gg d$, where $d$ is the separation between the charges), the dipole behaves like a point dipole. In this case, the general expression for the potential simplifies to:

This is a monopole-like potential that falls off as $1/r^2$, where the potential is proportional to the dipole moment $p$ and inversely proportional to the square of the distance from the dipole.

4. Electric Potential Due to a Dipole in Terms of Spherical Coordinates

In spherical coordinates ($r$, $\theta$), where $r$ is the radial distance from the dipole and $\theta$ is the angle with respect to the dipole axis, the potential due to a dipole is:

Here:

• $r$ is the distance from the origin (where the dipole is located),

• $\theta$ is the angle between the position vector and the dipole axis.

• At the axial points: $\theta = 0$, so $V(r, 0) = \frac{1}{4 \pi \epsilon_0} \frac{p}{r^2}$, which corresponds to the potential along the axis of the dipole.

• At the equatorial points: $\theta = \frac{\pi}{2}$, so $V(r, \frac{\pi}{2}) = 0$, meaning that the potential on the equatorial line of the dipole is zero, as expected from symmetry.

5. Potential Due to a Dipole in a Uniform Electric Field

A dipole placed in a uniform external electric field experiences a torque, which tends to align the dipole with the field. The potential energy $U$ of a dipole in a uniform electric field $\vec{E}$ is given by:

This expression gives the potential energy of the dipole in the external field, where:

• $\vec{p}$ is the dipole moment,
• $\vec{E}$ is the external electric field.

This potential energy is the work done by the external field to align the dipole with the field direction.

6. Conclusion

The electric potential due to a dipole has the following characteristics:

• It is scalar and depends on both the distance from the dipole and the angle with respect to the dipole axis.
• Axial line: The potential along the axis of the dipole falls off as $\frac{1}{r^2}$ and is positive for a positive dipole moment.
• Equatorial line: The potential along the equatorial line falls off as $\frac{1}{r^2}$ and is negative for a positive dipole moment.
• The general expression for the potential due to a dipole is $V(\vec{r}) = \frac{1}{4 \pi \epsilon_0} \frac{\vec{p} \cdot \hat{r}}{r^2}$.
• For points far from the dipole, the potential behaves as $1/r^2$, characteristic of a dipole.

This knowledge is important for understanding phenomena like molecular interactions, dipole-dipole interactions, and the behavior of dipoles in electric fields.