Voltage

Electric Potential

• Electric potential (or simply potential) is the electric potential energy per unit charge.

• $V=\frac{{\mathrm{PE}}_a}{q}$

  • $V$ is the electric potential (in $\mathsf{V}$, volts)
  • ${\mathrm{PE}}_a$ is the electric potential energy at point $a$ (in $\mathsf{J}$, joules)
  • $q$ is the charge (in $\mathsf{C}$, coulombs)

• $V=k\frac{Q}{r}$ (Coulomb potential)todo

  • $V$ is the electric potential due to a point charge $Q$ (in $\mathsf{V}$)
  • $Q$ is the point charge creating the electric potential (in $\mathsf{C}$)
  • $r$ is the distance between the charge and the point in space (in $\mathsf{m}$)
  • $k$ is Coulomb’s constant

Voltage

• Voltage (or (electrical) potential difference) is the difference in electric potential between two points in a circuit.

  • $V=V_a-V_b=\frac{{\mathrm{PE}}_a-{\mathrm{PE}}_b}{q}$ is the voltage between points $a$ and $b$.
    • $V$ is the work done in moving unit charge from $b$ to $a$ (in $\mathsf{V}$)
    • $V_a$ and $V_b$ are the electric potentials at points $a$ and $b$ (in $\mathsf{V}$)
  • The SI unit of voltage is the volt (V), defined as $1\mathsf{V}=1J/C$.
  • We often use ground (0 V) or infinity as a reference point.
  • Given a point $a$ with a higher potential $V_a$ and a point $b$ with a lower potential $V_b$:
    • A negative charge placed at $b$ has higher potential energy than at $a$, so it will move from $b$ to $a$ (when released) and decrease its potential energy.
      • ${\mathrm{PE}}_a<{\mathrm{PE}}_b⟹\Delta \mathrm{PE}<0$
    • A positive charge placed at $a$ has higher potential energy than at $b$, so it will move from $a$ to $b$ (when released) and decrease its potential energy.
      • ${\mathrm{PE}}_a>{\mathrm{PE}}_b⟹\Delta \mathrm{PE}>0$
    • In both cases, $\Delta V=V_a-V_b>0$.

• $V_{ba}=-dE$ is the electric field (uniform $\vec{\mathbf{E}}$)

  • $V_{ba}$ is the potential difference between points $a$ and $b$ (in $\mathsf{V}$)
  • $d$ is the distance between the points (in $\mathsf{m}$)
  • $E$ is the electric field (in $N/C$)

Electromotive Force

• todo A source of electromotive force (emf) (or source) is a device that transforms some other form of energy into electrical energy
• The potential difference (voltage) between the terminals of a source when no current is flowing is called the emf of the source
  • The emf of a source is determined by the chemical reactions that occur within the source
• The terminal voltage (difference) is the potential difference between the terminals of a source
• The internal resistance of a source is the resistance that the source itself has to the flow of current
  • Unless stated otherwise, we assume the battery’s internal resistance is negligible, and the battery voltage given is its terminal voltage
• $V=\mathcal{E}-Ir$ is the terminal voltage of a source
  • $\mathcal{E}$ is the emf of the source (in $\mathsf{V}$)
  • $I$ is the current that flows through the source (in $\mathsf{A}$)
  • $r$ is the internal resistance of the source (in $\mathsf{\Omega }$)
  • When $I=0$ (no current is flowing), $V=\mathcal{E}$ (the terminal voltage equals the emf)