Voltage

Electric Potential

• The electric potential (or potential) at a point is the amount of electric potential energy per unit charge at that point. $V=\frac{U}{q}$

• The electric potential at a point is the work done in moving a unit positive charge from infinity to that point.

• The volt ($\mathsf{V}$), defined as $1\mathsf{V}=1J/C$, is the SI unit of electric potential

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

  • $V$ is the electric potential at a point in space 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

• The electric potential at a point due to multiple charges is the sum of the potentials due to each charge. $V_{\text{total}}=V_1+V_2+V_3+...$

• todo

  • $V=-\int \vec{\mathbf{E}}\cdot d\vec{\mathbf{r}}$
  • $\Delta V=V_b-V_a=-{\int }_a^b\vec{\mathbf{E}}\cdot d\vec{\mathbf{r}}$
  • $V=k\int \frac{dq}{r}$

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 ($\mathsf{V}$)
  • 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, $\Delta V=V_a-V_b>0$.

• $V_{ba}=-dE$

  • $V_{ba}$ 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 $N/C$) (uniform $\vec{\mathbf{E}}$)

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)