Electrostatics – Equipotential Lines & Electric Dipoles

Equipotential Lines

  • Definition: A curve/surface where electrical potential is the same at every point ➜ \Delta V = 0 between any two points on the same line.
    • In drawings: appear as concentric circles around a point charge.
    • In 3-D reality: these are concentric spheres.
  • Work considerations
    • Moving test charge q{test} along one equipotential line ⇒ W = q{test}\,\Delta V = 0 (no work).
    • Moving between two different equipotential lines ⇒ W = q{test}\,(Vf-V_i).
    • Path-independence: work depends only on potential difference (conservative field).
  • Analogy: Sliding an object horizontally on a level surface (gravitational PE unchanged because height is unchanged; only vertical displacement matters).

Example: Electron from A to B near a huge +Q

  • Several paths (curvy, straight, etc.) are shown.
  • Least work? All paths require the SAME work because W depends solely on \Delta V.
  • Sign details
    • Source charge +Q ⇒ point B is lower electric potential than point A.
    • Electron (−q) → electric potential energy U=qV becomes higher at B than at A (because q is negative), i.e., electron must gain energy to be moved farther from +Q.

Electric Dipoles

  • Composition: two equal & opposite charges +q and -q separated by distance d.
    • Can be transient (London dispersion) or permanent (water, carbonyl, etc.).
  • Mental image: barbell – weights represent ±q; bar length = d.

Electric Potential of a Dipole at Point P

  • Diagram labels
    • r_1 = distance from P to +q.
    • r_2 = distance from P to -q.
    • r = distance from P to dipole midpoint.
  • General potential (scalar superposition):
    V = k\frac{q}{r1} - k\frac{q}{r2} = kq\frac{r2-r1}{r1 r2}
  • Far-field ( r \gg d ) approximations:
    • r1 r2 \approx r^2
    • r2 - r1 \approx d\cos\theta (θ = angle between r and dipole axis).
  • Therefore
    V(\text{far}) = k\frac{q d \cos\theta}{r^2}
  • Dipole moment (vector): \mathbf{p} = q\,d (units: C·m)
    • Physicists: arrow from −q to +q.
    • Chemists: arrow from +q to −q (often with crosshatch on + end).
    • Scalar magnitude: p = q d.
  • Far-field potential in compact form:
    V = k\frac{p\cos\theta}{r^2}

Numerical Example: Water Molecule

  • Given: p{H2O}=1.85\,\text{D} (Debye);
    1\,\text{D}=3.34\times10^{-30}\,\text{C·m}
  • Point of interest: on the dipole axis, r = 89\,\text{nm}=89\times10^{-9}\,\text{m}
    • On axis ⇒ \theta = 0^\circ\Rightarrow \cos\theta=1
  • Convert dipole moment:
    p = 1.85\times3.34\times10^{-30}=6.17\times10^{-30}\,\text{C·m}
  • Coulomb constant: k = 8.99\times10^{9}\,\text{N·m}^2\text{/C}^2
  • Potential:
    V = k\frac{p}{r^2} = 8.99\times10^{9}\frac{6.17\times10^{-30}}{(89\times10^{-9})^2}
    \approx 7.01\times10^{-6}\,\text{V}
  • Rounded estimate in lecture: 6.7\times10^{-6}\,\text{V}

Special Equipotential: Perpendicular Bisector of a Dipole

  • Plane halfway between +q and −q; angle to dipole axis = 90^\circ.
  • Since \cos90^\circ = 0 ⇒ V = 0 everywhere on this plane.
  • Electric field magnitude on this plane (far-field approximation):
    E = \frac{1}{4\pi\varepsilon_0}\frac{p}{r^3}
  • Direction of \mathbf{E} along bisector: opposite to \mathbf{p} (physicist convention).

Torque on a Dipole in a Uniform External Field

  • In a uniform field \mathbf{E}:
    • Force on +q: \mathbf{F}_{+}=+q\mathbf{E}
    • Force on −q: \mathbf{F}_{-}=-q\mathbf{E}
    • Forces equal & opposite ⇒ translational equilibrium (net force = 0).
  • Torque about center (using lever arm d/2 for each charge):
    \tau = \left(\frac{d}{2}FE\sin\theta\right)+\left(\frac{d}{2}FE\sin\theta\right)=dF_E\sin\theta
    =d(qE)\sin\theta = pE\sin\theta
  • Compact formula: \boxed{\tau = pE\sin\theta}
    • p = magnitude of dipole moment.
    • E = magnitude of external uniform field.
    • \theta = angle between \mathbf{p} and \mathbf{E}.
  • Consequence: torque tends to rotate dipole so \mathbf{p} aligns with \mathbf{E} (stable equilibrium when vectors are parallel).

Concept Connections & Implications

  • Equipotential lines reinforce conservative nature of electrostatic fields (path-independent work), analogous to gravitational potential.
  • Dipole concepts bridge electrostatics to molecular chemistry (reaction mechanisms, intermolecular forces).
  • Torque on dipoles underlies dielectrophoresis, molecular alignment in external fields, and behavior of polar molecules in capacitors.
  • Mathematical approximations (far-field) crucial for simplifying complex charge distributions into effective dipoles in biology and material science.