Question

An electric dipole of dipole moment \(\mathbf{p}\), fixed in direction, is located at a position \(\mathbf{r}_{0}(t)\) with respect to the origin. Its velocity \(\mathbf{y}=d \mathbf{r}_{0} / d t\) is nonrelativistic. (a) Show that the dipole's charge and current densities can be expressed formally as $$ \rho(\mathbf{x}, t)=-(\mathbf{p} \cdot \nabla) \delta\left(\mathbf{x}-\mathbf{r}_{0}(t)\right) ; \quad \mathbf{J}(\mathbf{x}, t)=-\mathbf{v}(\mathbf{p} \cdot \mathbf{\nabla}) \delta\left(\mathbf{x}-\mathbf{r}_{0}(t)\right) $$ (b) Show that the off-center moving dipole gives rise to a magnetic dipole field and an electric quadrupole field in addition to an electric dipole field, with moments $$ m=\frac{1}{2} \mathbf{p} \times \mathbf{v} $$ and $$ Q_{u}=3\left(x_{0} p_{j}+x_{0} p_{i}\right)-2 \mathbf{r}_{0} \cdot \mathbf{p} \delta_{i j} $$ [There are, of course, still higher moments.] (c) Show that the quasi-static electric quadrupole field is $$ \mathbf{E}(\mathbf{x})=\frac{1}{4 \pi \epsilon_{0}} \frac{1}{r^{4}}\left[15 \mathbf{n}\left(\mathbf{n} \cdot \mathbf{r}_{0}\right)(\mathbf{n} \cdot \mathbf{p})-3 \mathbf{r}_{0}(\mathbf{n} \cdot \mathbf{p})-3 \mathbf{p}\left(\mathbf{n} \cdot \mathbf{r}_{0}\right)-3 \mathbf{n}\left(\mathbf{r}_{0}+\mathbf{p}\right)\right] $$ where \(\mathrm{n}\) is a unit vector in the radial direction.

Short answer

The dipole's motion modifies its charge and current densities, introduces a magnetic dipole and electric quadrupole moment, leading to a specified electric field.

Step-by-step solution

Expressing Charge Density

We start by examining the expression for the charge density of a dipole. For a dipole moment \( \mathbf{p} \) located at \( \mathbf{r}_0(t) \), the charge density \( \rho(\mathbf{x}, t) \) can be expressed using the relation: \( \rho(\mathbf{x}, t) = - (\mathbf{p} \cdot abla) \delta(\mathbf{x} - \mathbf{r}_0(t)) \). This is derived considering that the dipole can be modeled by two point charges whose separation tends to zero while the product with the charge magnitude tends to \( \mathbf{p} \). The gradient \( abla \) acts on the delta function, effectively differentiating it to reflect the influence of the dipole moment's components in space.

Expressing Current Density

Next, we look at the current density \( \mathbf{J}(\mathbf{x}, t) \). Using the velocity \( \mathbf{v} = d\mathbf{r}_0 / dt \), the current density can be written as \( \mathbf{J}(\mathbf{x}, t) = -\mathbf{v}(\mathbf{p} \cdot abla) \delta(\mathbf{x} - \mathbf{r}_0(t)) \). This accounts for the fact that the dipole is moving with velocity \( \mathbf{v} \), and thus the motion increases the effective current density as it involves both displacement of charge and the dipole moment.

Magnetic Dipole Moment

For a moving dipole, the magnetic dipole moment \( \mathbf{m} \) arises due to the relative motion between charges modeled by the dipole. The expression is given as \( \mathbf{m} = \frac{1}{2} \mathbf{p} \times \mathbf{v} \). This comes from the analogy with moving charges forming a current loop, where the dipole moment is seen as a loop with infinitesimal area, while its edge is described by motion \( \mathbf{v} \) with the dipole moment \( \mathbf{p} \).

Electric Quadrupole Moment

The electric quadrupole moment \( Q_{ij} \) is derived to characterize the field more accurately as the dipole moves off-center. The expression is given by \( Q_{ij} = 3(x_i p_j + x_j p_i) - 2 \mathbf{r}_0 \cdot \mathbf{p} \delta_{ij} \). This formulation reflects the spatial distribution of the dipole moment across coordinates, capturing the change in electric field due to the motion effectively creating a structure with quadrupole symmetry.

Electric Quadrupole Field

Finally, the quasi-static electric quadrupole field \( \mathbf{E}(\mathbf{x}) \) is expressed as: \[ \mathbf{E}(\mathbf{x}) = \frac{1}{4 \pi \epsilon_0} \frac{1}{r^4}\left[15 \mathbf{n}(\mathbf{n} \cdot \mathbf{r}_0)(\mathbf{n} \cdot \mathbf{p}) - 3 \mathbf{r}_0(\mathbf{n} \cdot \mathbf{p}) - 3 \mathbf{p}(\mathbf{n} \cdot \mathbf{r}_0) - 3 \mathbf{n}(\mathbf{r}_0 \cdot \mathbf{p})\right], \] where \( \mathbf{n} \) is the unit vector in radial direction. This expression captures how the electric field is influenced by the spatial distribution of the moving dipole and its alignment with respect to the radial direction from the origin.

Key concepts

Dipole Moment

An electric dipole consists of two equal and opposite charges separated by a distance. The dipole moment is a vector that characterizes the strength and direction of this separation. It is denoted as \( \mathbf{p} \) and calculated by the formula \( \mathbf{p} = q \cdot \mathbf{d} \), where \( q \) is the magnitude of the charge and \( \mathbf{d} \) is the displacement vector pointing from the negative to the positive charge.
The direction of the dipole moment is crucial in understanding how the dipole interacts with external fields. When placed in an electric field, a dipole experiences a torque that aligns it along the field lines. The strength of this interaction is directly proportional to the dipole moment itself.
Moreover, the dipole moment is instrumental in calculating fields produced by the dipole at various points in space. It helps determine how charged particles will be affected by the dipole's presence.

Charge Density

Charge density is a measure of electric charge per unit volume. When exploring electric dipoles, charge density \( \rho(\mathbf{x}, t) \) can be represented formally by considering how the dipole moment operates within the chosen space.
For a point dipole located at position \( \mathbf{r}_0(t) \), the charge density is expressed as:

• \( \rho(\mathbf{x}, t) = - (\mathbf{p} \cdot abla) \delta(\mathbf{x} - \mathbf{r}_0(t)) \)

Here, \( abla \) is the gradient operator, and \( \delta(\mathbf{x} - \mathbf{r}_0(t)) \) is the Dirac delta function, which shows that the charge density is focused at the dipole's location.
This expression aids in visualizing the distribution of charge relative to the dipole moment and captures the dipole's tendency to influence points near or at \( \mathbf{r}_0(t) \) in space.

Current Density

Current density relates to the amount of current flowing through a unit area in a material or space. Within the context of electric dipoles, current density \( \mathbf{J}(\mathbf{x}, t) \) becomes relevant when the dipole is in motion.
For a moving dipole with velocity \( \mathbf{v} \), the current density is described by:

• \( \mathbf{J}(\mathbf{x}, t) = -\mathbf{v}(\mathbf{p} \cdot abla) \delta(\mathbf{x} - \mathbf{r}_0(t)) \)

This formula showcases how the velocity of the dipole influences the distribution of current density around its position. The gradient \( abla \) and the Dirac delta function capture the spatial effect relative to the dipole's trajectory.
Understanding current density for moving dipoles is crucial when predicting how they interact with magnetic fields and provide insight into their dynamic properties.

Electric Quadrupole Field

An electric quadrupole field represents a more complex distribution of electric field lines compared to a simple dipole. It arises when a dipole moves off-center, adding a layer of spatial complexity to the electric field analysis.
The quadrupole field for a moving dipole is expressed as:

• \[ \mathbf{E}(\mathbf{x}) = \frac{1}{4 \pi \epsilon_{0}} \frac{1}{r^{4}} \left[15 \mathbf{n}(\mathbf{n} \cdot \mathbf{r}_{0})(\mathbf{n} \cdot \mathbf{p}) -3 \mathbf{r}_{0}(\mathbf{n} \cdot \mathbf{p}) - 3 \mathbf{p}(\mathbf{n} \cdot \mathbf{r}_{0}) - 3 \mathbf{n}(\mathbf{r}_{0} \cdot \mathbf{p}) \right] \]

This formula illustrates how the field's intensity and direction modifies as a function of \( \mathbf{r}_0 \) and \( \mathbf{p} \), with \( \mathbf{n} \) being the unit radial vector. Its complexity makes it vital for studying interactions in systems where electric fields are not uniform and dipoles cannot be considered static within space.
These quadrupole arrangements occur in many real-world scenarios, making them an important study area in electrostatics and fields.