Electric Fields and Electric Dipoles

A set of comprehensive vocabulary flashcards covering electric field fundamentals, superposition, electric dipoles and multipoles, continuous charge distributions, charged-particle dynamics, and dipole torque and energy.

Field force

A type of force that acts through space to produce a physical effect even when there is no direct physical contact between interacting bodies.

Two-step interaction model

A framework for understanding electromagnetic interactions where: (1) a source charge modifies the surrounding space by establishing an electric field, and (2) a secondary charge placed in that field interacts with it and experiences an electrostatic force.

Electric field (operational definition)

The vector quantity defined as the force experienced per unit charge by a positive test charge placed at that location: $\mathbf{E}(\mathbf{r}) = \frac{\mathbf{F}_{\text{on } q_0}}{q_0}$ (for $q_0 > 0$), with SI units of $\text{N/C}$ (or $\text{V/m}$).

Electric field of a point charge

The electric field produced by an isolated point source charge $Q$, given by $\mathbf{E} = k_e \frac{Q}{r^2} \hat{\mathbf{r}}$, where $k_e = \frac{1}{4\pi\varepsilon_0}$; the field points radially outward for positive charges ($Q > 0$) and radially inward for negative charges ($Q < 0$).

Principle of superposition (electric fields)

The principle stating that the total electric field at a point $P$ created by a collection of $N$ source charges is the vector sum of the individual electric fields: $\mathbf{E}(P) = \sum_{i=1}^N \mathbf{E}_i(P) = \frac{1}{4\pi\varepsilon_0} \sum_{i=1}^N \frac{q_i}{r_i^2}\hat{\mathbf{r}}_i$.

Electric dipole

A configuration consisting of a pair of equal and opposite charges, $+q$ and $-q$, separated by a distance vector $\mathbf{d}$.

Electric dipole moment ($\mathbf{p}$)

A vector quantity characterizing an electric dipole, defined as $\mathbf{p} = q\mathbf{d}$, with units of $\text{C}\cdot\text{m}$, pointing in the direction from the negative charge toward the positive charge.

Far-field dipole approximation (perpendicular bisector)

The limiting expression for the electric field of an electric dipole along its equatorial axis at distances $r \gg d$, given by $\mathbf{E}_\perp \approx -\frac{1}{4\pi\varepsilon_0} \frac{\mathbf{p}}{r^3}$, oriented opposite to the dipole moment $\mathbf{p}$.

Far-field dipole approximation (axial line)

The limiting expression for the electric field of an electric dipole along its central axis at distances $r \gg d$, given by $\mathbf{E}_\parallel \approx \frac{1}{4\pi\varepsilon_0} \frac{2\mathbf{p}}{r^3}$, oriented parallel to the dipole moment $\mathbf{p}$.

Electric quadrupole

A charge distribution that can be viewed as two oppositely directed dipoles separated by a small distance, possessing zero net charge and zero net dipole moment, whose leading axial electric field term scales as $\frac{1}{z^4}$.

Multipole distance hierarchy

The characteristic spatial power-law decay of electric fields with distance $r$ far from charge configurations: monopoles scale as $\frac{1}{r^2}$, dipoles scale as $\frac{1}{r^3}$, and quadrupoles scale as $\frac{1}{r^4}$.

Continuous superposition

The integral method used to determine the total electric field created by a continuous distribution of charge: $\mathbf{E} = k_e \int \frac{\hat{\mathbf{r}}}{r^2}\,dq$.

Linear charge density ($\lambda$)

The quantity of electric charge per unit length of a one-dimensional body, defined as $\lambda = \frac{dq}{d\ell}$, measured in units of $\text{C/m}$.

Surface charge density ($\sigma$)

The quantity of electric charge per unit surface area of a two-dimensional boundary or sheet, defined as $\sigma = \frac{dq}{dA}$, measured in units of $\text{C/m}^2$.

Volume charge density ($\rho$)

The quantity of electric charge per unit volume within a three-dimensional region, defined as $\rho = \frac{dq}{dV}$, measured in units of $\text{C/m}^3$.

Axial electric field of a uniformly charged rod

The net electric field along the symmetry axis of a uniformly charged rod of length $L$ carrying total charge $Q$, at a distance $a$ beyond one end: $E = k_e \frac{Q}{a(a + L)}$.

Axial electric field of a uniformly charged ring

The net electric field on the central axis at a distance $x$ from the center of a uniformly charged circular ring of radius $a$ carrying total charge $Q$: $E_x = k_e \frac{Qx}{(x^2 + a^2)^{3/2}}$.

Charged-particle acceleration in a uniform electric field

The constant acceleration imparted to a particle of mass $m$ and charge $q$ by a uniform field $\mathbf{E}$, given by $\mathbf{a} = \frac{q}{m}\mathbf{E}$, directed along $\mathbf{E}$ for positive charges and opposite $\mathbf{E}$ for negative charges.

Cathode-ray tube (CRT)

A specialized vacuum tube that directs an electron beam using an electron gun, shapes and steers it via transverse vertical and horizontal deflection plates, and projects it onto a fluorescent screen.

Torque on an electric dipole

The rotational couple exerted on an electric dipole $\mathbf{p}$ by an external electric field $\mathbf{E}$, given vectorially by $\boldsymbol{\tau} = \mathbf{p} \times \mathbf{E}$ with magnitude $\tau = pE\sin(\theta)$, tending to rotate $\mathbf{p}$ into alignment with $\mathbf{E}$.

Potential energy of an electric dipole

The orientation-dependent potential energy of a dipole $\mathbf{p}$ in an external field $\mathbf{E}$, defined as $U = -\mathbf{p} \cdot \mathbf{E} = -pE\cos(\theta)$, referenced to zero energy at $\theta = 90^\circ$.

Stable equilibrium (electric dipole)

The dipole orientation occurring at $\theta = 0$ where the dipole moment $\mathbf{p}$ aligns parallel to the field $\mathbf{E}$, yielding minimum potential energy ($U_{\min} = -pE$) and a restoring torque against small displacements.

Unstable equilibrium (electric dipole)

The dipole orientation occurring at $\theta = \pi$ ($180^\circ$) where the dipole moment $\mathbf{p}$ is antiparallel to the field $\mathbf{E}$, resulting in maximum potential energy ($U_{\max} = +pE$) where any slight rotation drives the dipole further from alignment.