Charge Sign Conventions

Charge signs are a common source of correct-looking but physically reversed formulas.

Default Convention

The symbol $e$ denotes the positive elementary charge magnitude:

$$e>0, \qquad q_e=-e.$$

For a particle of charge $q$, minimal coupling uses

$$\mathbf p\to\mathbf p-q\mathbf A, \qquad H=\frac{(\mathbf p-q\mathbf A)^2}{2m}+q\Phi.$$

Translation Table

Source Convention	Translate To Default
electron charge written as $e$ with $e<0$	replace by $q_e=-e_{\mathrm{positive}}$
minimal coupling written $\mathbf p+e\mathbf A$ for electron	same as $\mathbf p-q_e\mathbf A$ with $q_e=-e$
Coulomb attraction written $-e^2/(4\pi\epsilon_0r)$	$e$ is charge magnitude; sign is in the product of charges
magnetic moment written with electron $g$ factor	track both $q_e=-e$ and spin convention

Lorentz Force Check

For charge $q$,

$$\mathbf F=q(\mathbf E+\mathbf v\times\mathbf B).$$

An electron accelerates opposite to $\mathbf E$ because $q=-e$.

Common Mistakes

• Reading $e$ as the electron charge on one page and as a positive magnitude on another.
• Changing the sign of $\mathbf A$ instead of the particle charge.
• Forgetting that holes in condensed matter are effective positive charges.
• Mixing SI and Gaussian electromagnetic units while debugging a sign.

Canonical Links

• Units and Constants
• Minimal Coupling in Wave Mechanics
• Charged Particle in Magnetic Field
• Constants

References

• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• J. D. Jackson, Classical Electrodynamics, 3rd ed., Wiley, 1998.