C = Q / Delta V
Capacitance equals charge divided by potential difference.
Capacitance as the ratio of stored charge to potential difference.
- C
- Capacitance (F)
- Q
- Magnitude of charge on either conductor (C)
- Delta V
- Potential difference between the conductors (V)
Use when — Defining or computing capacitance for any two-conductor system in electrostatic equilibrium.
Common trap — Treating C as changing with Q for a fixed geometry and medium; C is constant for that system.
C = epsilon0 A / d
Capacitance equals permittivity of free space times area divided by separation.
Capacitance of a parallel-plate capacitor with vacuum or air between the plates.
- epsilon0
- Permittivity of free space (F/m)
- A
- Area of overlap of the plates (m^2)
- d
- Separation between the plates (m)
Use when — Plate separation is small compared to plate dimensions, so the field between the plates is uniform and fringing is neglected.
Common trap — Using the full plate area when only part of the plates overlap; use the overlapping area only.
C = K epsilon0 A / d
Capacitance equals dielectric constant times permittivity of free space times area divided by separation.
Capacitance of a parallel-plate capacitor completely filled with a dielectric of constant K.
- K
- Dielectric constant of the medium (dimensionless)
- epsilon0
- Permittivity of free space (F/m)
- A
- Area of overlap of the plates (m^2)
- d
- Separation between the plates (m)
Use when — The dielectric slab completely fills the gap between the plates and is the same thickness as the separation.
Common trap — Applying this directly when the dielectric only partially fills the gap; a partial slab needs a series combination of the filled and unfilled regions.
1/C_series = 1/C1 + 1/C2 + ...
Reciprocal of the series equivalent capacitance equals the sum of the reciprocals of the individual capacitances.
Equivalent capacitance of capacitors connected in series.
- C_series
- Equivalent series capacitance (F)
- C1, C2
- Individual capacitances (F)
Use when — Each capacitor in the chain carries the same charge and the connection has no branch points between them.
Common trap — Adding capacitances directly instead of adding reciprocals for a series path.
C_parallel = C1 + C2 + ...
The parallel equivalent capacitance equals the sum of the individual capacitances.
Equivalent capacitance of capacitors connected in parallel.
- C_parallel
- Equivalent parallel capacitance (F)
- C1, C2
- Individual capacitances (F)
Use when — Each capacitor shares the same pair of nodes and hence the same potential difference.
Common trap — Applying the series reciprocal rule to capacitors that actually share the same two nodes.
U = (1/2) C (Delta V)^2
Energy stored equals one half times capacitance times potential difference squared.
Energy stored in a capacitor expressed using capacitance and potential difference.
- U
- Electrostatic energy stored (J)
- C
- Capacitance (F)
- Delta V
- Potential difference across the capacitor (V)
Use when — The potential difference is fixed or known, such as when an ideal battery remains connected.
Common trap — Reusing the pre-change Delta V after the connection state has changed the voltage.
U = Q^2 / (2C)
Energy stored equals charge squared divided by two times capacitance.
Energy stored in a capacitor expressed using charge and capacitance.
- U
- Electrostatic energy stored (J)
- Q
- Charge on the capacitor (C)
- C
- Capacitance (F)
Use when — Charge is fixed or known, such as for an isolated capacitor.
Common trap — Using the original C after a dielectric or geometry change has already altered it.
u = (1/2) epsilon E^2
Energy density equals one half times permittivity times electric field squared.
Electrostatic energy stored per unit volume of the field region between the plates.
- u
- Energy density (J/m^3)
- epsilon
- Permittivity of the medium between the plates (F/m)
- E
- Electric field magnitude in the region (V/m)
Use when — Interpreting capacitor energy as stored in the field rather than only in the charge on the plates, for a uniform-field region such as between parallel plates.
Common trap — Using the vacuum permittivity when a dielectric fills the region; use the medium's permittivity.
Q1 + Q2 = (C1 + C2) V_common
The sum of the initial charges equals the sum of the capacitances times the common final potential.
Total charge is conserved when two charged capacitors are connected, giving a single common final potential.
- Q1, Q2
- Charges on the two capacitors before connection (C)
- C1, C2
- Capacitances of the two capacitors (F)
- V_common
- Common potential difference after connection (V)
Use when — Two previously charged capacitors are connected plate to plate and allowed to reach a shared potential.
Common trap — Assuming the final energy equals the initial total energy; energy is generally not conserved in this idealised redistribution.
Delta U = C1 C2 (V1 - V2)^2 / (2 (C1 + C2))
Energy lost equals the product of the two capacitances times the square of their initial potential difference, divided by twice their sum.
Energy lost when two capacitors at different initial potentials are connected together.
- Delta U
- Energy lost during redistribution (J)
- C1, C2
- Capacitances of the two capacitors (F)
- V1, V2
- Initial potential differences on the two capacitors (V)
Use when — Computing how much electrostatic energy is dissipated when two charged capacitors at different potentials are connected.
Common trap — Applying this when the two capacitors already start at the same potential; the loss is zero in that case.
q(t) = Q0 (1 - e^(-t / RC))
Charge at time t equals the final charge times one minus the exponential of negative t over R C.
Charge on a capacitor as a function of time while it charges through a resistor from a battery.
- q(t)
- Charge at time t (C)
- Q0
- Final steady charge (C)
- R
- Resistance in the charging path (ohm)
- C
- Capacitance (F)
- t
- Time elapsed since charging began (s)
Use when — A capacitor charges through a resistor from a constant-voltage source, treated as the capacitor-side result of that circuit.
Common trap — Extending this to a circuit with multiple resistors or branches without first reducing it to a single effective resistance and capacitance.
q(t) = Q0 e^(-t / RC)
Charge at time t equals the initial charge times the exponential of negative t over R C.
Charge on a capacitor as a function of time while it discharges through a resistor.
- q(t)
- Charge at time t (C)
- Q0
- Initial charge at the start of discharge (C)
- R
- Resistance in the discharge path (ohm)
- C
- Capacitance (F)
- t
- Time elapsed since discharging began (s)
Use when — A previously charged capacitor discharges through a resistor with no source in the loop.
Common trap — Forgetting that the time constant RC is the same quantity that governs both charging and discharging for the same resistor-capacitor pair.
