Simplifying Physics Dimensions and Units: A New Approach for Enhanced Understanding

Mathematics and physics are like a foreign language. To master a foreign language, you need to work on it daily, review it, use it, and see value in it. If you have a poor memory like mine, having well-organized and indexed notes can be incredibly helpful. This is especially true for the vast and complex nature of physics, where understanding the dimensions and units of different physical quantities can be particularly challenging.

Handwritten Notes

Hope it helps.

The Standard Dimension List’s Limitations

The standard list of dimensions often adds a lot of unnecessary complexity and fails to serve well, especially when it comes to understanding the sizes at different scales and the use of negative powers. I have rewritten the dimensions to overcome these limitations by changing the base units. For example, density is pretty constant from atoms to stars, while velocity is an indicator of how ‘hot’ a system is and remains relatively constant. Time, on the other hand, is the only truly independent thing.

A New Approach to Units and Dimensions

I have introduced a DC unit and length and time to represent the vast zoo of units one encounters in physics. By setting ( T 1 ), ( V 10 ), ( mu 100 ), and ( S 300 ) (where ( S ) is the surface density of charge which gives density ( D mu S^2 700 )), the units in each hundred can be expressed as follows: -1xx: Siemens -110 Farad -109 x: second 1 metre 11 metre/sec 10 metre/sec2 9 1xx: Ohm 110 Henry 111 3xx: Ampere 321 Coulomb 322 4xx: Volt 431 Weber 432 Tesla 410 7xx: Watt 752 Joule 753 kilogram 733 newton 742 and Pascal 720 The units can then add arithmetically. By creating a page for each row of the table with rows running in steps of 11 and columns in steps of 10, one usually needs 0, 10, 20, 30, and rows to 55. That is usually enough to handle the lot. The numbers simply add. Therefore, Ampere's law ( F qv times B ) simplifies to ( q C 322 ), ( v text{m/s} 11^{-1} 10 ), ( B T 410 ), and ( 322 times 10 times 410 742 N ).

Memorization Techniques

Many units will be memorized through repeated usage. Another technique is to remember a formula that contains a quantity whose units you want to know. The dimensions can be handled as algebraic quantities. By memorizing the basic SI units and constructing the other units from them, you can easily derive the units of complex equations. For example:

The magnetic induction field ( vec{B} ) has the unit Tesla (T). If you remember the magnetic force on a moving charge is ( vec{F} q vec{v} times vec{B} ), then in terms of units, you get:

N C (frac{m}{s}) T

Solving for T, you get:

T ( frac{Ns}{Cm} frac{N}{frac{C}{s}m} frac{N}{Am} )

The last equality requires remembering the definition of current as the rate of flow of charge.