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 $\vec{\mathbf{r}}$ is the work done in moving a unit positive charge from infinity to that point. $V=V(\vec{\mathbf{r}})-V(\infty )=-{\int }_{\infty }^{\vec{\mathbf{r}}}\vec{\mathbf{E}}\cdot d\vec{\mathbf{l}}$
• The volt ($\mathrm{V}$), defined as $1\mathrm{V}=1J/C$, 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 $\mathrm{V}$)
  • $Q$ is the point charge creating the electric potential (in $\mathrm{C}$)
  • $r$ is the distance between the charge and the point in space (in $\mathrm{m}$)
  • $k$ is Coulomb’s constant
• The electric potential at a point due to multiple charges is the sum of the potentials due to each charge. $V_{\text{total}}=V_1+V_2+V_3+...$

Voltage

$V=V_a-V_b=\frac{W_{a\to b}}{q}=-\frac{\Delta U}{q}={\int }_a^b\vec{\mathbf{E}}\cdot d\vec{\mathbf{l}}=-{\int }_b^a\vec{\mathbf{E}}\cdot d\vec{\mathbf{l}}$is the voltage (or (electrical) potential difference) between points $a$ and $b$. (also called the potential difference of $a$ with respect to $b$)

• $V_a$ and $V_b$ are the electric potentials at points $a$ and $b$ (in $\mathrm{V}$)
• $\Delta U=(U_b-U_a)$
• $\frac{W_{a\to b}}{q}=-\frac{\Delta U}{q}$ is the work done by the electric field in moving a unit charge $q$ from point $a$ to point $b$ (in $\mathrm{J}$)
• The SI unit of voltage is the volt
• If $V_a>V_b$:
  • A negative charge $q^{-}$ placed at $b$ has higher potential energy than at $a$ (i.e. $U_a<U_b$)
  • A positive charge $q^{+}$ placed at $b$ has lower potential energy than at $a$ (i.e. $U_a>U_b$)
  • In both cases, $V=V_a-V_b>0$
• We often use ground (0 V) or infinity as a reference point.

Electromotive Force

• A source (of emf) is a device that transforms some other form of energy into electrical energy
• The potential difference (voltage) be tween 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 $\mathrm{V}$)
  • $I$ is the current that flows through the source (in $\mathrm{A}$)
  • $r$ is the internal resistance of the source (in $\mathrm{\Omega }$)
  • When $I=0$ (no current is flowing), $V=\mathcal{E}$ (the terminal voltage equals the emf)