Write Every Quantity in M, L and T, and Let Homogeneity Do the Rest
Dimensions are a bookkeeping device: every quantity can be written as a product of powers of mass, length and time. Two rules do most of the work. Only quantities of the same dimensions can be added or equated, and a constant that appears with a quantity takes the dimensions that make the equation consistent.
Units and Dimensions of Physical Quantities is the central idea in this question pattern. Identify the condition in the stem before applying the method below.
Common dimensions and the homogeneity rule
[F] = M L T⁻² and [Energy] = M L² T⁻².
Since 1/sqrt(μ0 ε0) = c, it has the dimensions of velocity, L T⁻¹. Angle, strain, relative density and refractive index have no dimensions, but Planck's constant and energy do not share dimensions, so they differ.
Changing units and base quantities
A unit of energy is a unit of force times a unit of length, so if the unit of length and the unit of force are each made 4 times larger, the unit of energy becomes 16 times larger. If energy E, velocity V and time T are chosen as base quantities, then mass = E V⁻² and surface tension (force per length, M T⁻²) is E V⁻² T⁻²: write the target as E^a V^b T^c and equate the powers of M, L and T.
Units of constants in an equation
To find the unit of a constant in a field equation, divide the field by the rest of each term. For E = A/x³ i + B y j + C z² k, A has the unit of E x³ which is N m³/C, B has the unit of E/y which is V/m² (since E is in V/m), and C has the unit of E/z² which is N/(m² C).
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- ✓[F] = M L T⁻²; [E] = M L² T⁻²; [h] = M L² T⁻¹; [G] = M⁻¹ L³ T⁻²
- ✓Quantities added or equated must have the same dimensions (homogeneity)
- ✓Dimensionless: angle, strain, relative density, refractive index
- ✓1/sqrt(μ0 ε0) has the dimensions of velocity
- ✓A constant's unit = (unit of the term) / (unit of the variable it multiplies)
Do not treat Planck's constant and energy as the same dimension. Energy is h times frequency, so h = energy x time, one factor of time more than energy, which is why they cannot be interchanged in a dimensional check.
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