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:
  $V<em>{\text{total}} = \sum</em>i V<em>i = \sum</em>i k \frac{Q<em>i}{r</em>i}$.

Potential Difference (Voltage)

• If points $A$ and $B$ sit at different distances from a charge distribution, a potential difference (voltage) exists:
  $\Delta V = V<em>B - V</em>A$.
• 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).