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 $q<em>{test}$ along one equipotential line ⇒ $W = q</em>{test}\,\Delta V = 0$ (no work).
  • Moving between two different equipotential lines ⇒ $W = q<em>{test}\,(V</em>f-V_i)$.
  • 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):
  $V = k\frac{q}{r<em>1} - k\frac{q}{r</em>2} = kq\frac{r<em>2-r</em>1}{r<em>1 r</em>2}$
• Far-field ( $r \gg d$ ) approximations:
  • $r<em>1 r</em>2 \approx r^2$
  • $r<em>2 - r</em>1 \approx d\cos\theta$ (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: $p<em>{H</em>2O}=1.85\,\text{D}$ (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):
  $\tau = \left(\frac{d}{2}F<em>E\sin\theta\right)+\left(\frac{d}{2}F</em>E\sin\theta\right)=dF_E\sin\theta$
  $=d(qE)\sin\theta = pE\sin\theta$
• 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.