Potential & Voltage

• The electric potential $\phi$ is defined as the electric energy divided by the charge $q$ of a point charge inside an electric field:

$$\phi = \frac{E_\mathrm{pot}}{q}$$

• The potential difference between two points and thus the integral over the electric field is called voltage:

$$U = \phi_2 - \phi_1 = \int \vec{E}\cdot\mathrm{d}\vec{s}$$ Analog to the potential energy in mechanics, a similar concept can be applied to electrodynamics. Starting from the formula for the potential energy of a point at $R$ up to infinity, the potential energy of the electric field is given as $$E_\mathrm{pot} = q\int_R^\infty\vec{E}\cdot\mathrm{d}\vec{s}$$ We are now considering a point charge $q_2$ inside the electric field of another one with the value $q_1$. Inserting Coulomb's law leads to the potential energy of $q_1$ against infinity: $$E_\mathrm{pot} = \int_R^\infty\frac{q_1q_2}{4\pi\varepsilon_0r^2}\,\mathrm{d}r = \frac{q_1q_2}{4\pi R}$$ To create a value that is independent of $q_2$, the electrical potential $\phi$ was introduced. The potential of a point charge can then be written as $$\phi = \frac{Q}{4\pi R}$$ The index 1 has been omitted in the formula above and $q$ was changed to $Q$ for reasons of simplicity. This formula could have been also derived by using the electric field strength instead of the force by writing $$\phi = \frac{E_\mathrm{pot}}{q} = \int_R^\infty \vec{E}\cdot\mathrm{d}\vec{s}$$ If a charge is moved from point A to B, the potential energy against infinity is not very interesting. It is, therefore, useful to define a new physical quantity called electrical voltage as the potential difference between these two points: $$U = \Delta \phi = \int_B^\infty \vec{E}\cdot \mathrm{d}\vec{s} - \int_A^\infty\vec{E}\cdot \mathrm{d}\vec{s} $$ The voltage is usually denoted with the letter $U$ and gets the unit Volt [V]. Calculating these integrals explicitly leads to a cancellation of the upper limits and the following term remains for the definition of the electrical voltage. Typical voltages of daily life are $1.5\,\mathrm{V}$ in the case of zinc-carbon batteries or 110 resp. 230 V for the energy grid in various countries. In the case of a homogenous electrical field, the field strength $E$ is then simply given as the product of the electric field and the distance. $$U = Ed$$ From the voltage $U$, the potential energy $E_\mathrm{pot}$ can be computed by multiplying the value with the charge $q$. $$\boxed{E_\mathrm{pot} = qU}$$ Of course, the concept of energy conservation can also be applied. If a charge $q$ is accelerated due to the application of a homogenous $E$, the speed can be calculated by comparing the potential energy with the kinetic energy: $$qU = \frac{1}{2}mv^2$$ This leads to the following formula for the velocity of the electron: $$\boxed{v = \sqrt{\frac{2qU}{m}}}$$ However, this formula only applies to non-relativistic particles. If the speed increases to higher values of more than 10% of the speed of light, the total energy of the electron is given as its rest energy in addition to the potential energy in the field: $$E = m_0c^2 + qU$$ Inserting the correlation $$E = \frac{m_0c^2}{\sqrt{1 - \frac{v^2}{c^2}}}$$ results in $$\frac{m_0c^2}{\sqrt{1 - \frac{v^2}{c^2}}} = m_0c^2 + qU$$ Solving this formula for $v$ gives $$\boxed{v = c\sqrt{1-\left(\frac{1}{1+\frac{qU}{m_0c^2}}\right)}}$$ Since the inner part of the square root can never be larger than 1, the maximum achievable speed of the particle is the speed of light.

• Electron in Electric Field (++)
• Gauss Law of Electrostatics (+++)