22-Elec-B9 Electromagnetic Field, Transmission Lines, Antennas, and Radiation · May 2016

Question 1 of 8: Maxwell's Equations in Free Space Reduced to One Spatial Variable

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Question 1: Maxwell's Equations in Free Space Reduced to One Spatial Variable (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Free space: no free charge and no conduction current, so $\rho = 0$ and $\mathbf{J} = 0$, with $\varepsilon = \varepsilon_0 = 8.854\times10^{-12}\ \text{F/m}$ and $\mu = \mu_0 = 4\pi\times10^{-7}\ \text{H/m}$. The fields are allowed to vary with time and with one spatial coordinate only; take that coordinate to be $z$, so that $\partial/\partial x = \partial/\partial y = 0$ for every field component. The curl expansion is supplied on the paper.

Find. The four Maxwell equations written out in cartesian components under this restriction, and a physical reading of the reduced set.

Approach. Write the four free-space equations, annihilate every $x$- and $y$-derivative using the supplied curl expansion, then sort the surviving scalar equations into those that constrain the longitudinal ($z$) components and those that couple the transverse components in pairs.

1. Start from the free-space Maxwell set. With no sources anywhere, the four equations in the field pair $(\mathbf{E}, \mathbf{H})$ are $$\nabla\cdot\mathbf{E} = 0,\qquad \nabla\cdot\mathbf{H} = 0,$$ $$\nabla\times\mathbf{E} = -\mu_0\frac{\partial \mathbf{H}}{\partial t},\qquad \nabla\times\mathbf{H} = \varepsilon_0\frac{\partial \mathbf{E}}{\partial t}.$$ These are the general statements; everything that follows is the consequence of one geometrical restriction imposed on them.
2. Impose the single spatial variable on the two divergence equations. With $\partial/\partial x = \partial/\partial y = 0$, the divergence collapses to its third term alone: $$\frac{\partial E_z}{\partial z} = 0, \qquad \frac{\partial H_z}{\partial z} = 0.$$ The longitudinal components therefore cannot vary along the one direction in which anything is allowed to vary.
3. Expand the two curl equations component by component. Substituting $\partial/\partial x = \partial/\partial y = 0$ into the aid leaves only the $\partial/\partial z$ terms, so $\operatorname{curl}(X,Y,Z)$ reduces to $\left(-\partial Y/\partial z,\ \partial X/\partial z,\ 0\right)$. Faraday's law then gives $$-\frac{\partial E_y}{\partial z} = -\mu_0\frac{\partial H_x}{\partial t}, \qquad \frac{\partial E_x}{\partial z} = -\mu_0\frac{\partial H_y}{\partial t}, \qquad 0 = -\mu_0\frac{\partial H_z}{\partial t},$$ and the Ampere–Maxwell law gives $$-\frac{\partial H_y}{\partial z} = \varepsilon_0\frac{\partial E_x}{\partial t}, \qquad \frac{\partial H_x}{\partial z} = \varepsilon_0\frac{\partial E_y}{\partial t}, \qquad 0 = \varepsilon_0\frac{\partial E_z}{\partial t}.$$
4. Dispose of the longitudinal components. The third component of each curl equation says $\partial H_z/\partial t = 0$ and $\partial E_z/\partial t = 0$; combined with Step 2, $E_z$ and $H_z$ are constant in both $z$ and $t$. A constant field carries no wave and no energy flow along $z$, so for any radiation field $$\boxed{E_z = H_z = 0}$$ and the field is purely transverse — a TEM field. This is a result, not an assumption: the one-variable restriction forces it.
5. Collect the four survivors into two independent pairs. Reading the remaining equations together, $E_x$ couples only to $H_y$, and $E_y$ couples only to $H_x$: $$\frac{\partial E_x}{\partial z} = -\mu_0\frac{\partial H_y}{\partial t}, \qquad \frac{\partial H_y}{\partial z} = -\varepsilon_0\frac{\partial E_x}{\partial t},$$ $$\frac{\partial E_y}{\partial z} = \mu_0\frac{\partial H_x}{\partial t}, \qquad \frac{\partial H_x}{\partial z} = \varepsilon_0\frac{\partial E_y}{\partial t}.$$ The two pairs are completely decoupled, which is exactly the statement that the two orthogonal linear polarisations propagate independently.
6. Eliminate one member of a pair to expose the wave equation. Differentiating the first pair's first equation with respect to $z$ and the second with respect to $t$, then substituting, $$\frac{\partial^2 E_x}{\partial z^2} = \mu_0\varepsilon_0\frac{\partial^2 E_x}{\partial t^2}.$$ This is the one-dimensional wave equation, whose general solution is $E_x = f(z - vt) + g(z + vt)$: an arbitrary forward wave plus an arbitrary backward wave, neither changing shape. The speed follows by inspection, $$\boxed{v_p = \frac{1}{\sqrt{\mu_0\varepsilon_0}} = 2.998\times10^{8}\ \text{m/s} = c}$$
7. Extract the ratio of the field magnitudes. Putting a single forward wave $E_x = f(z - ct)$ back into $\partial E_x/\partial z = -\mu_0\,\partial H_y/\partial t$ and integrating gives $E_x = \mu_0 c\,H_y$, so $$\boxed{\frac{E_x}{H_y} = \sqrt{\frac{\mu_0}{\varepsilon_0}} = \eta_0 = 376.7\ \Omega}$$ The electric and magnetic fields of a travelling wave are not independent: fixing one fixes the other, in phase and in a fixed ratio.

Comment on the results. Restricting free-space Maxwell to one spatial variable turns a set of eight coupled scalar equations into two independent pairs describing uniform plane waves. Three features are worth stating explicitly. First, the field is strictly transverse — no component survives along the direction of variation, which is why a plane wave in an unbounded medium has no longitudinal field and why a waveguide, which does support longitudinal fields, must depend on more than one coordinate. Second, the surfaces of constant phase are the planes $z = \text{constant}$, and the field is uniform over each of them, so the wave carries no beam spreading and no attenuation; this is an idealisation that a real field approaches only locally, far from its source. Third, the wave is non-dispersive: $v_p$ contains no frequency, so an arbitrary pulse shape propagates undistorted — the property that makes free space the reference against which every guided or material medium is compared.