A problem that cannot be solved on paper can often be solved in the laboratory. However, there are instances where a full-scale experiment
would be too costly. One such problem is that of the design of an MHD generator (Sec. 22.1).
In such cases it is sometimes useful to perform experiments on a model of convenient size. One then has the real system, for which the variables are $x, y, z, t, \boldsymbol{E}, \boldsymbol{B}, \epsilon_{r}, \mu_{r}, \sigma, R, C, L$, etc., and the model, whose variables are $x^{\prime}, y^{\prime}, z^{\prime}$, etc. The ratios $x / x^{\prime}, y / y^{\prime}, z / z^{\prime}$, etc. are the scale factors. Not all these factors can be chosen arbitrarily because both the unprimed and the primed variables must satisfy Maxwell's equations. The number of arbitrary scale factors is equal to 4 , the number of fundamental units (meter, kilogram, second, ampere). With mechanical systems there are only three arbitrary scale factors.
Let us set
$$
\frac{x}{x^{\prime}}=\frac{y}{y^{\prime}}=\frac{z}{z^{\prime}}=l, \quad \frac{t}{t^{\prime}}=\tau, \quad \frac{E}{E^{\prime}}=e, \quad \frac{H}{H^{\prime}}=h
$$
The other scale factors follow from Maxwell's equations and from other relations.
(a) Use Maxwell's equations to show that
(i) $\frac{\mu_{r}}{\mu_{r}^{\prime}}=\frac{\epsilon \tau}{l h}$,
(ii) $\frac{\epsilon_{r}}{\epsilon_{r}^{\prime}}=\frac{\tau h}{e l}$,
(iii) $\frac{\sigma}{\sigma^{\prime}}=\frac{h}{e l}$.
(b) Show that
(i) $\frac{\mathscr{S}}{\mathscr{S}^{\prime}}=e h$,
(ii) $\frac{J}{J^{\prime}}=\frac{h}{l}$,
(iii) $\frac{I}{I^{\prime}}=h l$,
(iv) $\frac{R}{R^{\prime}}=\frac{e}{h}$,
(v) $\frac{V}{V^{\prime}}=e l$.
(c) Show that
(i) $\frac{\Phi}{\Phi^{\prime}}=e \tau l$,
(ii) $\frac{B}{B^{\prime}}=\frac{e \tau}{l}$,
(iii) $\frac{C}{C^{\prime}}=\tau \frac{h}{\epsilon}$,
(iv) $\frac{L}{L^{\prime}}=\frac{e \tau}{h}$.
(d) Show that
(i) $\frac{f}{f^{\prime}}=\frac{1}{\tau}$,
(ii) $\frac{Q}{Q^{\prime}}=h l \tau$,
(iii) $\frac{\rho}{\rho^{\prime}}=\frac{\tau h}{l^{2}}$,
(iv) $\frac{P}{P^{\prime}}=h e l^{2}$.
(e) Show that, if $L$ is a length, then
$$
\frac{f \mu \sigma L^{2}}{f^{\prime} \mu^{\prime} \sigma^{\prime} L^{\prime 2}}=1
$$
(f) One author states that
$$
\frac{f^{2} \epsilon \mu}{f^{\prime 2} \epsilon^{\prime} \mu^{\prime}}=1
$$
Is he right?
In practice, these relations are simplified by the exclusion of ferromagnetic materials because of their nonlinearity. Then $\mu_{r} / \mu_{r}^{\prime}=1, e \tau=l h$, and there are only three independent scale factors. Then $\sigma / \sigma^{\prime}=\tau / l^{2}$ and $\sigma \omega L^{2}=\sigma^{\prime} \omega^{\prime} L^{\prime 2}$, where $L$ is a length.
If the fields are in a vacuum, $\epsilon_{r} / \epsilon_{r}^{\prime}=1$ and $\tau h=e l .$ Then $e=h$ and $\tau=1$. Note that if $\epsilon_{r} / \epsilon_{r}^{\prime}=1, \mu_{r} / \mu_{r}^{\prime}=1$, and $\sigma / \sigma_{r}=1$, then
$$
\frac{\tau h}{e l}=\frac{e \tau}{l h}=\frac{h}{e l}=1
$$
and $e=h, \tau=1, l=1 .$ Then the model is the same size as the original! If the model is to be a different size, then either $\epsilon_{r}$, or $\sigma$, or both, must be different. This condition is often impossible to satisfy.
At low frequencies one can disregard the displacement current, and hence attribute any value to the ratio $\epsilon_{r} / \epsilon_{r}^{\prime}$, or to $\tau \mathrm{h} / \mathrm{el}$.