Home / Equation sheets / Physics C: E&M equation sheet AP Physics C: Electricity and Magnetism
The Physics C: E&M equation sheet, annotated for 2026 Print this sheet Transcribed from the equations table of the official AP Physics C: Electricity and Magnetism Course and Exam Description (Effective Fall 2024). College Board provides this sheet on both sections of the exam, so nothing here needs memorizing; knowing when each equation applies is the skill the exam actually tests. The geometry and math reference tables printed alongside the equations are not reproduced here.
For the units these equations belong to, see the Physics C: E&M course hub .
Constants and Conversion Factors k = 1 4 π ε 0 = 9.0 × 10 9 ( N ⋅ m 2 ) / C 2 k = \frac{1}{4\pi\varepsilon_0} = 9.0 \times 10^9\ (\text{N} \cdot \text{m}^2)/\text{C}^2 k = 4 π ε 0 1 = 9.0 × 1 0 9 ( N ⋅ m 2 ) / C 2 Coulomb constant
ε 0 = 8.85 × 10 − 12 C 2 / ( N ⋅ m 2 ) \varepsilon_0 = 8.85 \times 10^{-12}\ \text{C}^2/(\text{N} \cdot \text{m}^2) ε 0 = 8.85 × 1 0 − 12 C 2 / ( N ⋅ m 2 ) Vacuum permittivity
μ 0 = 4 π × 10 − 7 ( T ⋅ m ) / A \mu_0 = 4\pi \times 10^{-7}\ (\text{T} \cdot \text{m})/\text{A} μ 0 = 4 π × 1 0 − 7 ( T ⋅ m ) / A Vacuum permeability
m p = 1.67 × 10 − 27 kg m_p = 1.67 \times 10^{-27}\ \text{kg} m p = 1.67 × 1 0 − 27 kg Proton mass
m n = 1.67 × 10 − 27 kg m_n = 1.67 \times 10^{-27}\ \text{kg} m n = 1.67 × 1 0 − 27 kg Neutron mass
m e = 9.11 × 10 − 31 kg m_e = 9.11 \times 10^{-31}\ \text{kg} m e = 9.11 × 1 0 − 31 kg Electron mass
e = 1.60 × 10 − 19 C e = 1.60 \times 10^{-19}\ \text{C} e = 1.60 × 1 0 − 19 C Elementary charge
1 eV = 1.60 × 10 − 19 J 1\ \text{eV} = 1.60 \times 10^{-19}\ \text{J} 1 eV = 1.60 × 1 0 − 19 J 1 electron volt
c = 3.00 × 10 8 m/s c = 3.00 \times 10^8\ \text{m/s} c = 3.00 × 1 0 8 m/s Speed of light
1 u = 1.66 × 10 − 27 kg = 931 MeV / c 2 1\ \text{u} = 1.66 \times 10^{-27}\ \text{kg} = 931\ \text{MeV}/c^2 1 u = 1.66 × 1 0 − 27 kg = 931 MeV / c 2 1 unified atomic mass unit
G = 6.67 × 10 − 11 m 3 / ( kg ⋅ s 2 ) = 6.67 × 10 − 11 N ⋅ m 2 / kg 2 G = 6.67 \times 10^{-11}\ \text{m}^3/(\text{kg} \cdot \text{s}^2) = 6.67 \times 10^{-11}\ \text{N} \cdot \text{m}^2/\text{kg}^2 G = 6.67 × 1 0 − 11 m 3 / ( kg ⋅ s 2 ) = 6.67 × 1 0 − 11 N ⋅ m 2 / kg 2 Universal gravitational constant
g = 9.8 m/s 2 g = 9.8\ \text{m/s}^2 g = 9.8 m/s 2 Magnitude of the acceleration due to gravity at Earth's surface
g = 9.8 N/kg g = 9.8\ \text{N/kg} g = 9.8 N/kg Magnitude of the gravitational field strength at Earth's surface
Electricity and Magnetism ∣ F ⃗ E ∣ = 1 4 π ε 0 ∣ q 1 q 2 ∣ r 2 = k ∣ q 1 q 2 ∣ r 2 \left| \vec{F}_E \right| = \frac{1}{4\pi\varepsilon_0} \frac{|q_1 q_2|}{r^2} = k \frac{|q_1 q_2|}{r^2} F E = 4 π ε 0 1 r 2 ∣ q 1 q 2 ∣ = k r 2 ∣ q 1 q 2 ∣ E ⃗ = F ⃗ E q \vec{E} = \frac{\vec{F}_E}{q} E = q F E E ⃗ = 1 4 π ε 0 ∫ d q r 2 r ^ \vec{E} = \frac{1}{4\pi\varepsilon_0} \int \frac{dq}{r^2} \hat{r} E = 4 π ε 0 1 ∫ r 2 d q r ^ Φ E = ∫ E ⃗ ⋅ d A ⃗ \Phi_E = \int \vec{E} \cdot d\vec{A} Φ E = ∫ E ⋅ d A ∮ E ⃗ ⋅ d A ⃗ = q enc ε 0 \oint \vec{E} \cdot d\vec{A} = \frac{q_{\text{enc}}}{\varepsilon_0} ∮ E ⋅ d A = ε 0 q enc Q total = ∫ ρ ( r ) d V Q_{\text{total}} = \int \rho(r)\, dV Q total = ∫ ρ ( r ) d V U E = 1 4 π ε 0 q 1 q 2 r U_E = \frac{1}{4\pi\varepsilon_0} \frac{q_1 q_2}{r} U E = 4 π ε 0 1 r q 1 q 2 V = 1 4 π ε 0 ∫ d q r V = \frac{1}{4\pi\varepsilon_0} \int \frac{dq}{r} V = 4 π ε 0 1 ∫ r d q Δ V = − ∫ a b E ⃗ ⋅ d r ⃗ \Delta V = -\int_a^b \vec{E} \cdot d\vec{r} Δ V = − ∫ a b E ⋅ d r E x = − d V d x E_x = -\frac{dV}{dx} E x = − d x d V Δ U E = q Δ V \Delta U_E = q \Delta V Δ U E = q Δ V C = Q Δ V C = \frac{Q}{\Delta V} C = Δ V Q C = κ ε 0 A d C = \frac{\kappa \varepsilon_0 A}{d} C = d κ ε 0 A U C = 1 2 Q Δ V U_C = \frac{1}{2} Q \Delta V U C = 2 1 Q Δ V κ = ε ε 0 \kappa = \frac{\varepsilon}{\varepsilon_0} κ = ε 0 ε I = d q d t I = \frac{dq}{dt} I = d t d q I = ∫ J ⃗ ⋅ d A ⃗ I = \int \vec{J} \cdot d\vec{A} I = ∫ J ⋅ d A E ⃗ = ρ J ⃗ \vec{E} = \rho \vec{J} E = ρ J R = ρ ℓ A R = \frac{\rho \ell}{A} R = A ρ ℓ I = Δ V R I = \frac{\Delta V}{R} I = R Δ V R eq,s = ∑ i R i R_{\text{eq,s}} = \sum_i R_i R eq,s = i ∑ R i 1 R eq,p = ∑ i 1 R i \frac{1}{R_{\text{eq,p}}} = \sum_i \frac{1}{R_i} R eq,p 1 = i ∑ R i 1 1 C eq,s = ∑ i 1 C i \frac{1}{C_{\text{eq,s}}} = \sum_i \frac{1}{C_i} C eq,s 1 = i ∑ C i 1 C eq,p = ∑ i C i C_{\text{eq,p}} = \sum_i C_i C eq,p = i ∑ C i τ = R eq C eq \tau = R_{\text{eq}} C_{\text{eq}} τ = R eq C eq ∮ B ⃗ ⋅ d A ⃗ = 0 \oint \vec{B} \cdot d\vec{A} = 0 ∮ B ⋅ d A = 0 F ⃗ B = q v ⃗ × B ⃗ \vec{F}_B = q\, \vec{v} \times \vec{B} F B = q v × B d B ⃗ = μ 0 4 π I d ℓ ⃗ × r ^ r 2 d\vec{B} = \frac{\mu_0}{4\pi} \frac{I\, d\vec{\ell} \times \hat{r}}{r^2} d B = 4 π μ 0 r 2 I d ℓ × r ^ F ⃗ B = ∫ I d ℓ ⃗ × B ⃗ \vec{F}_B = \int I\, d\vec{\ell} \times \vec{B} F B = ∫ I d ℓ × B ∮ B ⃗ ⋅ d ℓ ⃗ = μ 0 I enc \oint \vec{B} \cdot d\vec{\ell} = \mu_0 I_{\text{enc}} ∮ B ⋅ d ℓ = μ 0 I enc B sol = μ 0 n I B_{\text{sol}} = \mu_0 n I B sol = μ 0 n I Φ B = ∫ B ⃗ ⋅ d A ⃗ \Phi_B = \int \vec{B} \cdot d\vec{A} Φ B = ∫ B ⋅ d A E = ∮ E ⃗ ⋅ d ℓ ⃗ = − d Φ B d t \mathcal{E} = \oint \vec{E} \cdot d\vec{\ell} = -\frac{d\Phi_B}{dt} E = ∮ E ⋅ d ℓ = − d t d Φ B emf, printed as script E on the official sheet
∣ E sol ∣ = N ∣ d Φ B d t ∣ \left| \mathcal{E}_{\text{sol}} \right| = N \left| \frac{d\Phi_B}{dt} \right| ∣ E sol ∣ = N d t d Φ B L sol = μ core N 2 A ℓ L_{\text{sol}} = \frac{\mu_{\text{core}} N^2 A}{\ell} L sol = ℓ μ core N 2 A U L = 1 2 L I 2 U_L = \frac{1}{2} L I^2 U L = 2 1 L I 2 E = − L d I d t \mathcal{E} = -L \frac{dI}{dt} E = − L d t d I τ = L R eq \tau = \frac{L}{R_{\text{eq}}} τ = R eq L ω L C = 1 L C \omega_{LC} = \frac{1}{\sqrt{LC}} ω L C = L C 1 Mechanics v x = v x 0 + a x t v_x = v_{x0} + a_x t v x = v x 0 + a x t x = x 0 + v x 0 t + 1 2 a x t 2 x = x_0 + v_{x0} t + \frac{1}{2} a_x t^2 x = x 0 + v x 0 t + 2 1 a x t 2 v x 2 = v x 0 2 + 2 a x ( x − x 0 ) v_x^2 = v_{x0}^2 + 2 a_x (x - x_0) v x 2 = v x 0 2 + 2 a x ( x − x 0 ) Δ x = ∫ v x ( t ) d t \Delta x = \int v_x(t)\, dt Δ x = ∫ v x ( t ) d t Δ v x = ∫ a x ( t ) d t \Delta v_x = \int a_x(t)\, dt Δ v x = ∫ a x ( t ) d t x ⃗ cm = ∑ m i x ⃗ i ∑ m i \vec{x}_{\text{cm}} = \frac{\sum m_i \vec{x}_i}{\sum m_i} x cm = ∑ m i ∑ m i x i r ⃗ cm = ∫ r ⃗ d m ∫ d m \vec{r}_{\text{cm}} = \frac{\int \vec{r}\, dm}{\int dm} r cm = ∫ d m ∫ r d m λ = d d ℓ m ( ℓ ) \lambda = \frac{d}{d\ell} m(\ell) λ = d ℓ d m ( ℓ ) a ⃗ sys = ∑ F ⃗ m sys = F ⃗ net m sys \vec{a}_{\text{sys}} = \frac{\sum \vec{F}}{m_{\text{sys}}} = \frac{\vec{F}_{\text{net}}}{m_{\text{sys}}} a sys = m sys ∑ F = m sys F net ∣ F ⃗ g ∣ = G m 1 m 2 r 2 \left| \vec{F}_g \right| = \frac{G m_1 m_2}{r^2} F g = r 2 G m 1 m 2 ∣ F ⃗ f ∣ ≤ ∣ μ F ⃗ N ∣ \left| \vec{F}_f \right| \leq \left| \mu \vec{F}_N \right| F f ≤ μ F N F ⃗ s = − k Δ x ⃗ \vec{F}_s = -k \Delta \vec{x} F s = − k Δ x a c = v 2 r = r ω 2 a_c = \frac{v^2}{r} = r \omega^2 a c = r v 2 = r ω 2 K = 1 2 m v 2 K = \frac{1}{2} m v^2 K = 2 1 m v 2 W = ∫ a b F ⃗ ⋅ d r ⃗ W = \int_a^b \vec{F} \cdot d\vec{r} W = ∫ a b F ⋅ d r Δ K = ∑ W i = ∑ F ∥ , i d i \Delta K = \sum W_i = \sum F_{\parallel, i}\, d_i Δ K = ∑ W i = ∑ F ∥ , i d i Δ U = − ∫ a b F ⃗ cf ( r ) ⋅ d r ⃗ \Delta U = -\int_a^b \vec{F}_{\text{cf}}(r) \cdot d\vec{r} Δ U = − ∫ a b F cf ( r ) ⋅ d r F x = − d U ( x ) d x F_x = -\frac{dU(x)}{dx} F x = − d x d U ( x ) U s = 1 2 k ( Δ x ) 2 U_s = \frac{1}{2} k (\Delta x)^2 U s = 2 1 k ( Δ x ) 2 U G = − G m 1 m 2 r U_G = -\frac{G m_1 m_2}{r} U G = − r G m 1 m 2 Δ U g = m g Δ y \Delta U_g = m g \Delta y Δ U g = m g Δ y P avg = W Δ t = Δ E Δ t P_{\text{avg}} = \frac{W}{\Delta t} = \frac{\Delta E}{\Delta t} P avg = Δ t W = Δ t Δ E P inst = d W d t P_{\text{inst}} = \frac{dW}{dt} P inst = d t d W p ⃗ = m v ⃗ \vec{p} = m \vec{v} p = m v F ⃗ net = d p ⃗ d t \vec{F}_{\text{net}} = \frac{d\vec{p}}{dt} F net = d t d p J ⃗ = ∫ t 1 t 2 F ⃗ net ( t ) d t = Δ p ⃗ \vec{J} = \int_{t_1}^{t_2} \vec{F}_{\text{net}}(t)\, dt = \Delta \vec{p} J = ∫ t 1 t 2 F net ( t ) d t = Δ p v ⃗ cm = ∑ p ⃗ i ∑ m i = ∑ m i v ⃗ i ∑ m i \vec{v}_{\text{cm}} = \frac{\sum \vec{p}_i}{\sum m_i} = \frac{\sum m_i \vec{v}_i}{\sum m_i} v cm = ∑ m i ∑ p i = ∑ m i ∑ m i v i ω = d θ d t \omega = \frac{d\theta}{dt} ω = d t d θ α = d ω d t \alpha = \frac{d\omega}{dt} α = d t d ω ω = ω 0 + α t \omega = \omega_0 + \alpha t ω = ω 0 + α t θ = θ 0 + ω 0 t + 1 2 α t 2 \theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2 θ = θ 0 + ω 0 t + 2 1 α t 2 ω 2 = ω 0 2 + 2 α ( θ − θ 0 ) \omega^2 = \omega_0^2 + 2 \alpha (\theta - \theta_0) ω 2 = ω 0 2 + 2 α ( θ − θ 0 ) a r = r α a_r = r \alpha a r = r α τ ⃗ = r ⃗ × F ⃗ \vec{\tau} = \vec{r} \times \vec{F} τ = r × F I tot = ∑ I i = ∑ m i r i 2 I_{\text{tot}} = \sum I_i = \sum m_i r_i^2 I tot = ∑ I i = ∑ m i r i 2 I = ∫ r 2 d m I = \int r^2\, dm I = ∫ r 2 d m I ′ = I cm + M d 2 I' = I_{\text{cm}} + M d^2 I ′ = I cm + M d 2 α sys = ∑ τ I sys = τ net I sys \alpha_{\text{sys}} = \frac{\sum \tau}{I_{\text{sys}}} = \frac{\tau_{\text{net}}}{I_{\text{sys}}} α sys = I sys ∑ τ = I sys τ net K rot = 1 2 I ω 2 K_{\text{rot}} = \frac{1}{2} I \omega^2 K rot = 2 1 I ω 2 W = ∫ τ ⋅ d θ W = \int \tau \cdot d\theta W = ∫ τ ⋅ d θ L ⃗ = r ⃗ × p ⃗ = I ω ⃗ \vec{L} = \vec{r} \times \vec{p} = I \vec{\omega} L = r × p = I ω Δ L = ∫ τ d t \Delta L = \int \tau\, dt Δ L = ∫ τ d t Δ x cm = r Δ θ \Delta x_{\text{cm}} = r \Delta\theta Δ x cm = r Δ θ T = 2 π ω = 1 f T = \frac{2\pi}{\omega} = \frac{1}{f} T = ω 2 π = f 1 T s = 2 π m k T_s = 2\pi \sqrt{\frac{m}{k}} T s = 2 π k m T p = 2 π ℓ g T_p = 2\pi \sqrt{\frac{\ell}{g}} T p = 2 π g ℓ T phys = 2 π I m g d T_{\text{phys}} = 2\pi \sqrt{\frac{I}{m g d}} T phys = 2 π m g d I x = x max cos ( ω t + ϕ ) x = x_{\text{max}} \cos(\omega t + \phi) x = x max cos ( ω t + ϕ ) The other sheets