Electrostatics Cheatsheet

Cheat Codes & Shortcuts

• Coulomb’s Law: \( F = k \frac{|q_1 q_2|}{r^2} \), where \( k \approx 8.99 \times 10^9 \, \text{N·m}^2/\text{C}^2 \).
• Electric Field: \( E = \frac{F}{q} = k \frac{q}{r^2} \), direction depends on charge sign.
• Electric Potential: \( V = k \frac{q}{r} \), potential energy \( U = qV \).
• Gauss’s Law: \( \Phi_E = \oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\epsilon_0} \), where \( \epsilon_0 \approx 8.85 \times 10^{-12} \, \text{C}^2/\text{N·m}^2 \).
• Capacitance: \( C = \frac{Q}{V} \), parallel plate: \( C = \epsilon_0 \frac{A}{d} \).
• Electric Field of a Dipole: At axial point, \( E \approx \frac{2kp}{r^3} \), where \( p = qd \).
• Potential Energy: \( U = -\vec{p} \cdot \vec{E} \) for a dipole in an electric field.
• Conductors: Electric field inside a conductor is zero in electrostatic equilibrium.
• Superposition: Total \( \vec{E} \) or \( V \) is the vector/scalar sum of contributions from all charges.
• Electric Flux: \( \Phi_E = \vec{E} \cdot \vec{A} \) for uniform field and area.

Quick Reference Table

Concept	Formula	Description
Coulomb’s Law	\( F = k \frac{|q_1 q_2|}{r^2} \)	Force between two point charges.
Electric Field	\( E = k \frac{q}{r^2} \)	Field due to a point charge.
Electric Potential	\( V = k \frac{q}{r} \)	Potential due to a point charge.
Gauss’s Law	\( \oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\epsilon_0} \)	Relates electric flux to enclosed charge.
Capacitance	\( C = \epsilon_0 \frac{A}{d} \)	Capacitance of a parallel plate capacitor.
Dipole Field	\( E \approx \frac{2kp}{r^3} \)	Electric field along dipole axis.

Advice

Electrostatics Quick Tips

• Coulomb’s Law: Use \( F = k \frac{q_1 q_2}{r^2} \) for point charges; direction is along the line joining charges.
• Electric Field: \( E = \frac{F}{q} \), vector sum for multiple charges.
• Gauss’s Law: Apply for high-symmetry cases to simplify field calculations.
• Potential Energy: For multiple charges, sum pairwise energies: \( U = \sum k \frac{q_i q_j}{r_{ij}} \).
• Capacitors: Combine in series (\( \frac{1}{C_{\text{eq}}} = \sum \frac{1}{C_i} \)) or parallel (\( C_{\text{eq}} = \sum C_i \)).

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