Electric Potential & Potential Difference

Definition of Electric Potential

  • Electric potential (symbol V) quantifies the “electric potential energy per unit charge” at a point in an electric field.
  • Formal definition:
    • V = \frac{U}{q}
    • U: electric potential energy (J)
    • q: magnitude of the test charge (C)
  • Unit: volt (V).
    • 1\,\text{V} = 1\,\text{J}\,/\,\text{C}

Relation to Electric Potential Energy

  • Though the names sound similar, electric potential and electric potential energy refer to different, yet related, quantities:
    • U is the energy a charge possesses because of its position in an electric field.
    • V is the energy per unit charge, a property of the location itself, independent of whether a test charge is present.
  • Expressed mathematically once again: U = qV.

Electric Potential of a Point Charge

  • For a point in space at a distance r from a single source charge Q:
    • Electric potential energy: U = k \frac{Qq}{r}.
    • Divide by q to isolate V (no test charge needed):
      V = k \frac{Q}{r}.
  • k is Coulomb’s constant, k \approx 8.99 \times 10^9 \, \text{N·m}^2/\text{C}^2.
  • V is a scalar; its sign is determined solely by the sign of the source charge Q:
    • Q > 0 \Rightarrow V > 0.
    • Q < 0 \Rightarrow V < 0.

Superposition for Multiple Charges

  • In a system with many charges, the total potential at any point is the scalar sum of individual potentials:
    V{\text{total}} = \sumi Vi = \sumi k \frac{Qi}{ri}.

Potential Difference (Voltage)

  • If points A and B sit at different distances from a charge distribution, a potential difference (voltage) exists:
    \Delta V = VB - VA.
  • Alternate formalism using work: \Delta V = \frac{W_{AB}}{q} where
    • W_{AB}: work needed to move a test charge q from A to B.
  • Work–potential link highlights a key property: work depends only on endpoints, not on the path—evidence that the electrostatic force is conservative.

Conservative‐Force Analogy

  • Like gravity, the electrostatic force conserves mechanical energy.
    • Path independence: any closed loop through a static electric field requires zero net work.

Spontaneous Motion & Sign Conventions

  • Charged particles naturally move toward lower potential energy. Whether this means lower or higher electric potential depends on the particle’s sign.

Positive Test Charge (q>0)

  • Spontaneously travels from higher V to lower V.
  • \Delta V = VB - VA < 0 (negative voltage experienced).
  • With q positive, W_{AB} = q\Delta V < 0—its electric potential energy decreases.

Negative Test Charge (q<0)

  • Spontaneously travels from lower V to higher V.
  • \Delta V = VB - VA > 0 (positive voltage experienced).
  • Because q is negative, W_{AB} = q\Delta V < 0 again—potential energy still decreases.

Key Takeaways

  • V = \frac{U}{q} is the core link between potential and potential energy.
  • For a point charge: V = kQ/r reveals the 1/r dependence.
  • Potential is a scalar; superposition applies via simple addition.
  • Voltage \Delta V equates to work per unit charge; electrostatic work is path‐independent, confirming a conservative field.
  • Direction of spontaneous motion:
    • Positive charges drift toward regions of lower V (negative \Delta V).
    • Negative charges drift toward regions of higher V (positive \Delta V).
  • In both cases, natural motion lowers electric potential energy (negative work done by the field).