Check the dimensional correctness of C = ε₀A/d
Parallel plate capacitor. The capacitance of two plates of area A a distance d apart.
Equation checker
Matching dimensions are necessary, not sufficient. A check cannot see pure numbers such as 1/2 or 2π, so a consistent equation can still be wrong.
C = ε0 A / dC[M-1 L-2 T4 A2]=epsilon0 A / d[M-1 L-2 T4 A2]Step by step
1. What each symbol stands for
- C capacitance [M-1 L-2 T4 A2]
- ε0 permittivity of free space [M-1 L-3 T4 A2]
- A area [L2]
- d length [L]
2. The dimensions of each term
- C [M-1 L-2 T4 A2] [M-1 L-2 T4 A2]
- epsilon0 A / d [M-1 L-3 T4 A2] × [L2] / [L] [M-1 L-2 T4 A2]
3. Compare
Every term has dimensions [M-1 L-2 T4 A2], so the equation is dimensionally consistent.
What it proves and what it does not
It can show
- Every term that is added or subtracted has the same dimensions, so the sum makes sense.
- Both sides of the equals sign have the same dimensions.
- Angles inside sin, cos and similar functions are pure numbers.
It cannot show
- A pure number such as 1/2 or 2π has no dimensions, so a check cannot see a missing or wrong one.
- Two quantities with the same dimensions, such as work and torque, cannot be told apart.
- A sign error or a wrong power that keeps the dimensions equal goes unnoticed.
Matching dimensions are necessary but not sufficient. Pure numbers such as 1/2 or 2π have no dimensions, so a check cannot see them, and an equation with matching dimensions can still be wrong.
Look up each quantity
C = ε₀A/d: quick answers
Is C = ε₀A/d dimensionally correct?
Yes. Every term has the same dimensions, [M⁻¹ L⁻² T⁴ A²], so C = ε₀A/d is dimensionally consistent. That does not prove the equation itself is correct.
Does a dimension check prove C = ε₀A/d is right?
No. A check can show that an equation is wrong, but it cannot show that it is right. Pure numbers such as 1/2 and 2π have no dimensions, so the check cannot see them.