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A LevelPhysicsCIERevision Notes1. Physical Quantities & Units1.2 SI UnitsHomogeneity of Physical Equations & Powers of Ten

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First teaching 2023

First exams 2025

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Homogeneity of Physical Equations & Powers of Ten (CIE A Level Physics)

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Katie M

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Katie M

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Homogeneity of physical equations

  • An important skill is to be able to check the homogeneity of physical equations using the SI base units
  • The combined units on either side of the equation should be the same
  • To check the homogeneity of physical equations:
    • Check the units on both sides of an equation
    • Determine if they are equal
    • If they do not match, the equation will need to be adjusted

Worked example

The speed of sound v  in a gas is given by

v space equals space square root of fraction numerator gamma p over denominator rho end fraction end root

where p  is gas pressure and ρ  is gas density.

Show that γ has no units.

Answer:

Step 1: Determine the units on the left:

  • The only term on the left is speed, which has units of m s−1

Step 2: Apply the homogeneity of physical equations:

  • This equation describes the speed of sound waves and is therefore physical
    • This means it must be homogeneous, so the units on the left must be equal to the combined units on the right

Step 3: Determine the combined SI base units of pressure:

  • Pressure is defined as the force F (units of N or kg m s−2) per unit area (units of m2) with the equation:

P space equals space F over A

  • Written in terms of units:

Pa space equals space fraction numerator kg space straight m space straight s to the power of negative 2 end exponent over denominator straight m squared end fraction space equals space kg space straight m to the power of negative 1 end exponent space straight s to the power of negative 2 end exponent

Step 4: Determine the combined SI base units of density:

  • Density is defined as mass m (units of kg) per unit volume V (units of m3) with the equation:

rho space equals space m over V

  • Written in terms of units:

units space of space density space equals space kg over straight m cubed space equals space kg space straight m to the power of negative 3 end exponent

Step 5: Equate the units of both sides of the equation:

  • The units of γ will be labelled as G

straight m space straight s to the power of negative 1 end exponent space equals space square root of fraction numerator straight G space up diagonal strike kg space straight m to the power of negative 1 end exponent space straight s to the power of negative 2 end exponent over denominator up diagonal strike kg space straight m to the power of negative 3 end exponent end fraction end root space equals space square root of straight G space straight m squared space straight s to the power of negative 2 end exponent end root space equals space square root of straight G space straight m space straight s to the power of negative 1 end exponent

  • This shows us that the square root of G is equal to 1, so G is equal to 1
  • Therefore γ has no units - this is sometimes called being dimensionless

Examiner Tip

There were multiple ways of answering this - you could have rearranged the equation to make γ the subject and shown that the other side of the equation and no units, or you could have found that, without γ in the equation, the equation was homogenous.

Powers of ten

  • Physical quantities can span a huge range of values
  • For example, the diameter of an atom is about 10–10 m (0.0000000001 m), whereas the width of a galaxy may be about 1021 m (1000000000000000000000 m)
  • This is a difference of 31 powers of ten
  • Powers of ten are numbers that can be achieved by multiplying 10 times itself
  • It is useful to know the prefixes for certain powers of ten

          Powers of ten

Prefix Abbreviation Power of 10
Tera- T 1012
Giga- G 109
Mega- M 106
Kilo- k 103
Centi- c 10−2
Milli- m 10−3
Micro- μ 10−6
Nano- n 10−9
Pico- p 10−12

Examiner Tip

You will often see very large or very small numbers categorised by powers of ten, so it is very important you become familiar with these as getting these prefixes wrong is a very common exam mistake!

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    Katie M

    Author: Katie M

    Expertise: Physics

    Katie has always been passionate about the sciences, and completed a degree in Astrophysics at Sheffield University. She decided that she wanted to inspire other young people, so moved to Bristol to complete a PGCE in Secondary Science. She particularly loves creating fun and absorbing materials to help students achieve their exam potential.