AP Physics C equation sheet, explained

By Ashraf Bhura, Co-Founder, STEM-Tutor · Updated October 2026

AP Physics C uses calculus, and its equation sheet shows it. Here are the key Mechanics and E&M equations and how to use them well.

Calculus is the key difference

Many AP Physics C equations are written as derivatives and integrals, such as v = dx/dt and W = ∫F·dr. That lets you handle forces, fields and charges that change, not just constant ones. Download the current official sheet from AP Central.

Mechanics: motion and forces

• v = dx/dt, a = dv/dt: velocity and acceleration as derivatives.
• ΣF = dp/dt = ma: Newton's second law, including variable mass in its momentum form.
• F_drag = −bv style resistive forces lead to differential equations and exponential approach to terminal velocity.

Mechanics: energy, momentum and rotation

• W = ∫F·dr: work by a variable force.
• F = −dU/dx: force from a potential energy function.
• J = ∫F dt: impulse from a changing force.
• I = ∫r² dm: moment of inertia, plus the parallel axis theorem I = I_cm + Md².
• τ = r × F, L = Iω: torque and angular momentum.

Mechanics: oscillations and gravitation

• x = A cos(ωt + φ) with ω = √(k/m): simple harmonic motion.
• U_G = −Gm₁m₂/r: gravitational potential energy far from Earth.

E&M: fields and potential

• ∮E·dA = Q_enc/ε₀: Gauss's law, for symmetric charge distributions.
• ΔV = −∫E·dr and E = −dV/dr: field and potential.
• C = Q/ΔV, U_C = ½QΔV: capacitance and stored energy.

E&M: circuits, magnetism and induction

• RC circuits: charge and current change exponentially with time constant τ = RC.
• ∮B·dl = μ₀I_enc: Ampère's law.
• dB = (μ₀/4π)(I dl × r̂)/r²: the Biot-Savart law.
• ε = −dΦ_B/dt: Faraday's law, with Lenz's law giving the direction.
• LR circuits: current grows or decays with time constant τ = L/R.

How to use it in the exam

1. Decide whether the quantity is constant. If not, you probably need the calculus form.
2. For Gauss's and Ampère's laws, draw the symmetry and the surface or loop first.
3. Set up integrals carefully: limits and the variable of integration earn marks.
4. Practise derivations. Free-response questions often ask you to derive an expression, not just use one.

Worked examples with calculus

Example 1: velocity from position

x(t) = 4t³ − 2t (metres, seconds). Find the velocity and acceleration at t = 2 s.

• v = dx/dt = 12t² − 2, so v(2) = 46 m/s.
• a = dv/dt = 24t, so a(2) = 48 m/s².

Example 2: work by a variable force

A spring force F(x) = −kx with k = 200 N/m. Find the work done by the spring as it is stretched from 0 to 0.1 m.

• W = ∫₀^0.1 (−200x) dx = −100x² from 0 to 0.1 = −1 J. The spring does negative work; the person stretching it does +1 J.

Example 3: moment of inertia of a rod

Find I for a uniform rod of mass M and length L about one end.

• dm = (M/L)dx, so I = ∫₀^L x²(M/L)dx = (M/L)(L³/3) = ML²/3.
• About the centre it is ML²/12; the parallel axis theorem gives ML²/12 + M(L/2)² = ML²/3, a useful check.

Example 4: Gauss's law

Find the electric field a distance r from a long straight wire with charge per unit length λ.

• Choose a cylindrical Gaussian surface of radius r and length L around the wire.
• E(2πrL) = λL/ε₀, so E = λ/(2πε₀r).

Example 5: an RC circuit

A 10 µF capacitor charges through a 100 kΩ resistor from a 9 V battery. Find the time constant and the capacitor voltage after one time constant.

• τ = RC = 10⁵ × 10⁻⁵ = 1 s.
• V = 9(1 − e⁻¹) ≈ 9 × 0.632 ≈ 5.7 V.

Example 6: Faraday's law

A 50-turn coil of area 0.02 m² sits in a field that falls uniformly from 0.4 T to 0 in 0.1 s. Find the induced emf.

• |ε| = N ΔΦ/Δt = 50 × (0.4 × 0.02)/0.1 = 4 V. Lenz's law says the induced current opposes the decrease.

Common mistakes

• Forgetting to change the limits when substituting in an integral.
• Choosing a Gaussian surface that does not match the symmetry.
• Treating a variable force as constant and using W = Fd.
• Losing the minus sign in Faraday's law; it carries the direction information.

Practice questions

Try each one before opening the answer.

The key ideas in more depth

The Physics C sheet lists general relationships, not special cases. For example, it gives W = ∫F·dr rather than W = Fd. Your job is to set up the integral or derivative for the situation in front of you. Practising that setup, with correct limits and variables, is the most valuable preparation for the free-response section.

When a question gives a graph instead of a formula, use the geometric meanings: area under a force-position graph is work, area under a force-time graph is impulse, and the slope of a potential energy graph gives the force (with a minus sign).

Frequently asked questions

Are the Mechanics and E&M sheets combined?

The College Board provides equation information for each exam. Check the current versions on AP Central.

Should I memorise derivations?

Know how to derive the standard results, such as the field of a sphere or a wire, because questions often ask you to do exactly that.

Key takeaways

• Physics C equations use calculus to handle forces, fields and currents that change.
• Set up integrals with the right variable and limits before calculating.
• Gauss's and Ampère's laws need symmetry to be useful.
• RC and LR circuits change exponentially with time constants RC and L/R.
• Practise derivations: free-response questions often ask for them.

Quick check