240 Potential Difference
A comparison of potential between two points that is used to tell how much influence changes from one place to another.
Work per unit charge. The potential difference between two points is the work per unit charge to carry a test charge from one point to the other. This principle is used to assign a single number to a pair of points.
The potential difference as work per unit charge is
\[ V(b) - V(a) = \dfrac{W}{q} \]
where
- \(V(a)\) and \(V(b)\) are the potentials at the two points.
- \(W\) is the work to move charge \(q\) from \(a\) to \(b\).
- \(q\) is the test charge.
The line integral of \(\mathbf{E}\). The same difference is minus the line integral of the electric field. This principle is used to compute voltage from a known \(\mathbf{E}\).
The line-integral form is
\[ V(b) - V(a) = -\displaystyle\int_{a}^{b}\mathbf{E}\cdot d\mathbf{l} \]
where
- \(V\) is the electric potential.
- \(\mathbf{E}\) is the electric field.
- \(d\mathbf{l}\) is a displacement along the path.
Independence of the zero. The value does not depend on where the zero of potential is chosen. This principle is used to treat only differences as physically meaningful.
Current and kinetic energy. A potential difference drives current through a resistive path and changes the kinetic energy of a charge that moves between the points. This principle is used to analyze circuits and particle accelerators.
Ohm’s law is
\[ I = \dfrac{\Delta V}{R} \]
where
- \(I\) is the current.
- \(\Delta V\) is the potential difference.
- \(R\) is the resistance.
Capacitance. For a pair of conductors the stored charge is proportional to the potential difference. This principle is used to define capacitance.
The capacitor relation is
\[ Q = C\,\Delta V \]
where
- \(Q\) is the magnitude of the charge on each conductor.
- \(C\) is the capacitance.
- \(\Delta V\) is the potential difference.
Note: Also called voltage when the influence is electric.
240.1 References
- Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — work per unit charge; path integral; physical significance.
- Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — kinetic energy change; driving current.
- Shankar, R. Fundamentals of Physics II. Yale University Press, 2020. — battery emf and voltage; capacitors.
- Logan, J. D. A First Course in Differential Equations. Springer, 2015. — voltage drop as work to move charge.
- Jaffe, R. L., & Taylor, W. The Physics of Energy. Cambridge University Press, 2018. — voltage and stored charge.
- Park, D. Introduction to the Quantum Theory. Dover, 2005. — Josephson effect.