Electric Potential & Energy — Rapid Review

Electric Potential (Point Charge)

  • Scalar quantity: energy per unit charge (J/C = V)

  • Formula: VA=kQrV_A = k\frac{Q}{r}

  • Electric field: EA=kQr2E_A = k\frac{Q}{r^2} (vector, points away from +Q, toward −Q)

  • Superposition: total V=∑<em>ikQ</em>iriV = \sum<em>i k\frac{Q</em>i}{r_i}

  • Reference: V(∞)=0V(\infty)=0

Electric Potential (Uniform Field)

  • Potential difference: ΔV=−Ed\Delta V = -E d (d = displacement parallel to E)

  • Magnitude: E=VABdE = \frac{V_{AB}}{d}, direction from high → low V

  • Positive charge: moves high → low V (along E)

  • Negative charge: moves low → high V (opposite E)

Work & Potential Difference

  • Work moving charge q: W=−q (V<em>B−V</em>A)=−q ΔVW = -q\,(V<em>B - V</em>A) = -q\,\Delta V

  • Easy move (W > 0): gains electric PE

  • Hard move (W < 0): loses electric PE

Electric Potential Energy (2-Charges)

  • Stored work: U=kQ<em>1Q</em>2rU = k\frac{Q<em>1 Q</em>2}{r}

  • Change during motion: ΔPE=q ΔV=−W\Delta PE = q\,\Delta V = -W

  • Multiple charges: U=∑<em>iiQjrijU = \sum<em>{ii Qj}{r{ij}}

Energy–Speed Relation

  • For a charge q: 12m(vf2−vi2)=−q (Vf−Vi)=q ΔV\tfrac12 m(vf^2 - vi^2) = -q\,(Vf - Vi) = q\,\Delta V
    (used for escape / impact speed problems)

Battery & Voltage

  • Voltage = fixed potential difference: V=V+−V−=ΔPEqV = V+ - V- = \frac{\Delta PE}{q}

  • Acts as source of electric potential

Equipotential Lines & Surfaces

  • Always perpendicular to E-field lines

  • No work to move along an equipotential

  • Highest (most +) near +Q, lowest (most −) near −Q

  • Conducting surface: W=0W=0 to move charge anywhere on it (surface is equipotential)

Quick Formulas & Facts

  • Point charge: V=kQ/rV = kQ/r, E=kQ/r2E = kQ/r^2

  • Work: W=−q ΔVW = -q\,\Delta V

  • Energy: U=kQ1Q2/rU = kQ1 Q2 / r

  • Uniform field: E=ΔV/dE = \Delta V/d, ΔV=−Ed\Delta V = -E d

  • Positive q moves toward lower V; negative q toward higher V