Electric Potential Energy Study Notes

Overview

  • Electric potential energy (EPE) is a form of stored energy associated with the relative positions of electric charges.
  • It is analogous to other potential energies (gravitational, elastic, chemical) but involves electric forces produced by charges or electric fields.
  • Units: both energy and work share the SI unit “joule” (J).

Major Types of Potential Energy Mentioned

  • Gravitational potential energy
    • Depends on mass, height, and gravitational field (Ug=mghU_g = mgh close to Earth’s surface).
  • Elastic potential energy
    • Stored in deformed springs or elastic objects (Ue=12kx2U_e = \frac12 k x^2 for ideal springs).
  • Chemical potential energy
    • Stored in chemical bonds; released/absorbed in chemical reactions.
  • Electric potential energy (focus of the lecture)
    • Arises from Coulomb interactions between charges.

Fundamental Equation for Electric Potential Energy

  • Given two point charges QQ (source) and qq (test), separated by distance rr: U=kQqrU = k \dfrac{Q q}{r}
    • k = 8.99 \times 10^9\, \text{N·m}^2 \text{/C}^2 (electrostatic constant).
    • Q,qQ, q in coulombs (C), rr in meters (m).

Derivation & Relation to Work

  • Work needed to bring a test charge from infinity to a point in an electric field defines EPE:
    • Coulomb’s law: FE=kQqr2F_E = k \dfrac{Q q}{r^2}.
    • Assume radial motion (force ‖ displacement), so cos⁡θ=1\cos \theta = 1.
    • Differential work dW=FE drdW = F_E \, dr; integrating from ∞\infty to rr ⇒ W=U=kQqrW = U = k \dfrac{Q q}{r}.
    • Highlights that EPE is path-independent (conservative field).

Sign Conventions & Physical Meaning

  • Unlike charges (one +, one –)
    • Q q < 0 ⇒ U < 0.
    • More negative UU as rr decreases ⇒ system becomes more stable.
  • Like charges (both + or both –)
    • Q q > 0 ⇒ U > 0.
    • UU decreases (toward 0) as rr increases ⇒ stability increases with separation.
  • Visualization on a number line:
    • Moving left of zero (more negative) represents lower energy & higher stability for unlike charges.
    • Moving toward zero from positive side (smaller +) represents lower energy & higher stability for like charges.
  • Important contrast with gravity: gravitational forces are always attractive; electrostatic forces can be either attractive or repulsive, causing both positive and negative potential energies.

Stability Discussion

  • Opposite Charges
    • At close separation → strong attraction, deep potential well (large |negative| UU) ⇒ highly stable.
  • Like Charges
    • At close separation → strong repulsion, high positive UU ⇒ unstable configuration.
    • Stability improves as charges spread apart (potential energy drops).

Worked Example

“If a charge of +2e and a charge of –3e are separated by 3 nm, what is the potential energy of the system?”

Parameters:

  • Fundamental charge e=1.6×10−19 Ce = 1.6 \times 10^{-19} \text{ C}.
  • Q=+2e=2(1.6×10−19)=3.2×10−19 CQ = +2e = 2(1.6 \times 10^{-19}) = 3.2 \times 10^{-19} \text{ C}.
  • q=−3e=−4.8×10−19 Cq = -3e = -4.8 \times 10^{-19} \text{ C}.
  • r=3 nm=3×10−9 mr = 3 \text{ nm} = 3 \times 10^{-9} \text{ m}.

Computation:
U=kQqr=(8.99×109)(3.2×10−19)(−4.8×10−19)3×10−9U = k \frac{Q q}{r} = (8.99 \times 10^{9}) \frac{(3.2 \times 10^{-19})(-4.8 \times 10^{-19})}{3 \times 10^{-9}}

Step-by-step numerical handling:

  1. Product of charges: 3.2×4.8=15.36⇒15.36×10−38 C23.2 \times 4.8 = 15.36 \Rightarrow 15.36 \times 10^{-38} \text{ C}^2.
  2. Multiply by kk: 8.99×109×15.36×10−38=138.1×10−298.99 \times 10^{9} \times 15.36 \times 10^{-38} = 138.1 \times 10^{-29}.
  3. Divide by rr: 138.1×10−293×10−9≈46.0×10−20\frac{138.1 \times 10^{-29}}{3 \times 10^{-9}} \approx 46.0 \times 10^{-20}.
  4. Include negative sign from charge product: U≈−4.6×10−19 JU \approx -4.6 \times 10^{-19} \text{ J}.
  • Final answer matches transcript: U=−4.6×10−19 JU = -4.6 \times 10^{-19} \text{ J}.
  • Interpretation: Negative value ⇒ attractive configuration, system is more stable at this separation.

Connections & Real-World Relevance

  • EPE concept underlies:
    • Binding energy in atoms/molecules (electron–nucleus attractions, electron–electron repulsions).
    • Capacitor energy storage (U=12CV2)\left(U = \frac12 C V^2\right) derived from pairwise charge interactions.
    • Electric potential (voltage) definition V=Uq=kQrV = \frac{U}{q} = k \frac{Q}{r}, crucial for circuits and electrochemistry.
    • Macroscopic phenomena: lightning (charge separation potential energy released), electrostatic precipitators, etc.
  • Ethical/practical note: Proper management of high voltages prevents accidents; understanding EPE helps engineers safely design power systems.

Key Takeaways / Exam Tips

  • Memorize U=kQqrU = k \dfrac{Q q}{r} and sign rules.
  • Remember that more negative UU = more stable for unlike charges; smaller positive UU = more stable for like charges.
  • Work done by/against the field equals the change in EPE: ΔU=−Wfield\Delta U = -W_{\text{field}}.
  • Even though gravitational & electric potentials look similar mathematically, gravity never produces a positive potential energy for two masses.
  • Typical test questions involve:
    • Calculating EPE between two or more charges.
    • Relating EPE to electric potential (voltage).
    • Reasoning about stability or work required to move charges.

Practice deriving energy changes for multiple-charge systems by summing pairwise energies: U{\text{total}} = \sum\limits{i<j} k \dfrac{qi qj}{r_{ij}}.