Understanding Dimensionally Correct CGS Units: Clarifying Common Misconceptions

It's a common misconception to believe that some CGS units might be dimensionally incorrect. However, the truth is rather simpler and more nuanced. Let's delve into the concept of dimensional analysis and explain why such beliefs are based on a misunderstanding.

Dimensions in Physical Quantities

The key to understanding this concept lies in the nature of physical entities and their dimensions. Physical quantities, such as force or charge, inherently carry dimensions, but these dimensions are derived from the fundamental scales and units used to measure them.

Force and Dimensional Analysis

For example, the force F can be defined as the product of mass M and acceleration a, which gives us the dimensional formula ML/T2. However, we can define a separate scale for force based on the mass that it can lift, M. This leads to coherent CGS, MKS, and FPS units, each with their own specific scales for measuring force. In CGS, for instance, the unit of force is given as dyne, which is related to the meter, gram, and second units through these scales.

Gravitational Units and Dimensional Consistency

In more complex systems, such as gravitational units, the constant g (acceleration due to gravity) plays a crucial role. The dimension of g is T-2/L, which means that the dimension of force in CGS becomes Mg, where M and L are the units of mass and length, respectively. This results in a consistent dimensional formula: M(L/T2.

Charges and Currents in CGS and SI

The Gaussian system of units, used in conjunction with CGS, often defines the unit of charge based on the force between charges. The equation F Q2/L2 can be used to derive units of charge in CGS. Similarly, the unit of current can be defined as F/L I2/L, leading to F I2. This is a perfectly legitimate scale for measuring current, as it is consistent with the underlying dimensions.

Maxwell's Equations and Dimensional Analysis

Maxwell's equations, particularly when dealing with electromagnetic fields, require a more comprehensive approach to dimensional analysis. The Gaussian system uses LMT (length, mass, time) as its basic units, while SI uses LMTI (length, mass, time, electric current). For complete dimensional consistency, six basic units are required, each with its own numerical value in different systems.

Maxwell's Equation in Dimensional Form

Consider Maxwell's equation:

kappa nabla E dD/j beta

This equation maintains dimensional consistency, with κ and β representing specific constants in both CGS and SI. In SI, κ β 1, while in CGS, κ c, β 1/(4π). This highlights the importance of these constants in determining the dimensional correctness of the equation.

Conclusion

It's crucial to understand that physical quantities, whether defined in CGS or SI units, are dimensionally correct as long as the scales and units used are coherent. Misconceptions arise when the underlying scales and constants are not properly accounted for. By recognizing these scales and constants, we can ensure that our physical equations, including Maxwell's equations, maintain dimensional consistency across different systems of units.