Conversions Between Systems (Units: A Unit Conversion Program and Scientific Calculator)

#### 8.1.3 Conversions Between Different Systems ¶

The CGS systems define units that measure the same thing but may have conflicting dimensions. Furthermore, the dimensions of the electromagnetic CGS units are never compatible with SI. But if you measure charge in two different systems you have measured the same physical thing, so there is a correspondence between the units in the different systems, and `units` supports conversions between corresponding units. When running with SI, `units` defines all of the CGS units in terms of SI. When you select a CGS system, `units` defines the SI units and the other CGS system units in terms of the system you have selected.

    (Gaussian) You have: statA
               You want: abA
            * 3.335641e-11
            / 2.9979246e+10
    (Gaussian) You have: abA
               You want: sqrt(dyne)
    conformability error
            2.9979246e+10 sqrt_cm^3 sqrt_g / s^2
            1 sqrt_cm sqrt_g / s

In the above example, `units` converts between the current units statA and abA even though the abA, from the EMU system, has incompatible dimensions. This works because in Gaussian mode, the abA is defined in terms of the statA, so it does not have the correct definition for EMU; consequently, you cannot convert the abA to its EMU definition.

One challenge of conversion is that because the CGS system has fewer base units, quantities that have different dimensions in SI may have the same dimension in a CGS system. And yet, they may not have the same conversion factor. For example, the unit for the \({E}\) field and \({B}\) fields are the same in the Gaussian system, but the conversion factors to SI are quite different. This means that correct conversion is only possible if you keep track of what quantity is being measured. You cannot convert statV/cm to SI without indicating which type of field the unit measures. To aid in dimensional analysis, `units` defines various dimension units such as ‘`LENGTH`’, ‘`TIME`’, and ‘`CHARGE`’ to be the appropriate dimension in SI. The electromagnetic dimensions such as ‘`B_FIELD`’ or ‘`E_FIELD`’ may be useful aids both for conversion and dimensional analysis in CGS. You can convert them to or from CGS in order to perform SI conversions that in some cases will not work directly due to dimensional incompatibilities. This example shows how the Gaussian system uses the same units for all of the fields, but they all have different conversion factors with SI.

    (Gaussian) You have: statV/cm
               You want: E_FIELD
            * 29979.246
            / 3.335641e-05

    (Gaussian) You have: statV/cm
               You want: B_FIELD
            * 0.0001
            / 10000

    (Gaussian) You have: statV/cm
               You want: H_FIELD
            * 79.577472
            / 0.012566371

    (Gaussian) You have: statV/cm
               You want: D_FIELD
            * 2.6544187e-07
            / 3767303.1

The next example shows that the oersted cannot be converted directly to the SI unit of magnetic field, A/m, because the dimensions conflict. We cannot redefine the ampere to make this work because then it would not convert with the statampere. But you can still do this conversion as shown below.

    (Gaussian) You have: oersted
               You want: A/m
    conformability error
            1 sqrt_g / s sqrt_cm
            29979246 sqrt_cm sqrt_g / s^2
    (Gaussian) You have: oersted
               You want: H_FIELD
            * 79.577472
            / 0.012566371