Charge per volt gives capacitance, and from that ratio the dimensional formula of capacitance turns out to be [M⁻¹ L⁻² T⁴ I²], with negative powers on mass and length, a fourth power on time and a square on current.
A capacitor stores charge on two conductors, and its capacitance C = Q/V says how much charge it holds for each volt across it. Both Q and V have to be rewritten in base quantities before dividing, which is where the unusual powers come from. Below are the steps, the farad in base units, two capacitor checks, a worked parallel plate example and the slips. Series and parallel combinations are on the capacitance formula page.
Quick facts
Dimensional formula
[M⁻¹ L⁻² T⁴ I²]Defining formula
SI unit
farad (F) = C/VIn base units
kg⁻¹ m⁻² s⁴ A²Scalar or vector
ScalarChapter
Electrostatic potential and capacitance, Class 12
How to get the dimensional formula of capacitance
- Charge: Q = It, so Q is [IT].
- Voltage: V = W/Q, work [ML²T⁻²] divided by [IT], which is [ML²T⁻³I⁻¹].
- Capacitance: C = Q/V, so divide [IT] by [ML²T⁻³I⁻¹].
- Flip the voltage powers and add: M⁻¹, L⁻², T¹⁺³ = T⁴, I¹⁺¹ = I². Result [M⁻¹L⁻²T⁴I²].
The T⁴ is the part people forget. One T comes from the charge on top and three more from flipping the T⁻³ in voltage. The voltage page shows the middle step on its own.
The farad and its everyday sizes
In base units the bracket reads kg⁻¹ m⁻² s⁴ A². That is one farad (F). Put simply, a farad means a coulomb stored for each volt (C/V).
A farad is very large. Real capacitors are measured in microfarads (μF, 10⁻⁶ F), nanofarads (nF, 10⁻⁹ F) and picofarads (pF, 10⁻¹² F). The prefix changes the size, not the dimensions.
Numbers for a two plate capacitor
Two metal plates, each 0.010 m² in area, sit 1.0 mm apart in air. Find C and the energy stored at 12 V.
C = ε₀A/d = (8.854 × 10⁻¹²)(0.010) ÷ (1.0 × 10⁻³) = 8.85 × 10⁻¹¹ F, about 88.5 pF.
U = ½CV² = 0.5 × 8.85 × 10⁻¹¹ × 144 = 6.37 × 10⁻⁹ J.
Units: (F/m)(m²)/m = F, then F × V² = (C/V)V² = C V = J.
Every term used here can be checked with brackets. ε₀ is [M⁻¹L⁻³T⁴I²]; times L² over L gives [M⁻¹L⁻²T⁴I²], the capacitance bracket. The permittivity of free space page derives ε₀.
Energy and time constant checks
Check: energy stored U = ½CV²
Square the voltage bracket to get [M²L⁴T⁻⁶I⁻²], then bring in C at [M⁻¹L⁻²T⁴I²].
Adding exponents gives M¹, L², T⁻² and I⁰. The product is therefore [ML²T⁻²], an energy, as required.
Check: the time constant τ = RC
Resistance is [ML²T⁻³I⁻²]. Times capacitance, every power cancels except T: T⁻³⁺⁴ = T¹.
So RC is a time, which is why a capacitor charges over a timescale set by RC.
The resistance bracket is on the resistance page.
Matching formulas and slips
| Quantity | Formula |
|---|---|
| Capacitance of any capacitor | C = Q/V |
| Isolated sphere | C = 4πε₀R |
| Parallel plates | C = ε₀A/d |
| Capacitors in parallel | C₁ + C₂ |
- Writing T³ instead of T⁴. Remember the T from charge on top.
- Writing I¹. Current appears once in charge and once more from flipping voltage.
- Treating the farad as a base unit. It reduces to kg⁻¹ m⁻² s⁴ A².
- Mixing up capacitance with permittivity. ε₀ has L⁻³; C has L⁻².
Electrical constants and circuit quantities sit side by side under dimensional formulas for revision.
Frequently asked questions
In which unit is capacitance measured?
Farads (F). Each farad is C/V, or kg⁻¹ m⁻² s⁴ A² when broken down.
Why is there a T⁴ in the dimensions of capacitance?
Charge brings T¹, and voltage, which sits in the denominator, brings T⁻³. Dividing by T⁻³ adds three, giving T⁴.
Is capacitance a scalar?
Yes. Capacitance is a scalar that depends on the shape, size and spacing of the conductors and on the material between them.
How is the dimensional formula of capacitance related to permittivity?
Permittivity is capacitance per unit length. So ε₀ is [M⁻¹L⁻³T⁴I²], one more L⁻¹ than capacitance.
Does RC really have the dimensions of time?
Yes. Resistance [ML²T⁻³I⁻²] times capacitance [M⁻¹L⁻²T⁴I²] leaves [T], so RC is measured in seconds.