Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

Everything in this course will be written in one of three coordinate systems. As we have seen, each is completely characterised by its coordinates, its unit vectors and its three scale factors. The differential elements, and (as we shall see in Section 2.3.10) every differential operator, follow from the scale factors alone. Figure 2.3.25 shows one and the same point described in the three systems, and Table 2.3.4 collects the data.

One point P, three descriptions. Here P has \rho=2, z=1.5, \phi=75^\circ, hence r=\sqrt{2^2+1.5^2}=2.5 and \theta=\arctan(2/1.5)\approx53.1^\circ. (a) Cartesian (x,y,z): three lengths, and the unit vectors are the same at every point. — **Figure 2.3.25.** One point P, three descriptions. Here P has $ρ = 2$ , $z = 1.5$ , $ϕ = 7 5^{\circ}$ , hence $r = 2^{2} + 1. 5^{2} = 2.5$ and $θ = arctan (2/1.5) \approx 53. 1^{\circ}$ . (a) Cartesian $(x, y, z)$ : three lengths, and the unit vectors are the same at every point.

(b) Cylindrical (\rho,\phi,z): distance \rho from the z axis, the azimuth \phi and the height z. Here \vec a_\rho points away from the axis. — (b) Cylindrical $(ρ, ϕ, z)$ : distance $ρ$ from the $z$ axis, the azimuth $ϕ$ and the height $z$ . Here $a_{ρ}$ points away from the axis.

(c) Spherical (r,\theta,\phi): distance r from the origin and the two angles \theta, \phi, with \vec a_r pointing away from the origin. The vector \vec a_\phi is common to (b) and (c), and \vec a_z to (a) and (b). — (c) Spherical $(r, θ, ϕ)$ : distance $r$ from the origin and the two angles $θ$ , $ϕ$ , with $a_{r}$ pointing away from the origin. The vector $a_{ϕ}$ is common to (b) and (c), and $a_{z}$ to (a) and (b).

Table 2.3.4. The three orthogonal coordinate systems of the course side by side. The coordinates are listed in their right-handed order $(u_{1}, u_{2}, u_{3})$ , and $d S_{i}$ is the area element on the coordinate surface $u_{i} = const$ (its direction is $\pm a_{i}$ ).

	Cartesian	cylindrical	spherical
coordinates	$x, y, z$	$ρ, ϕ, z$	$r, θ, ϕ$
ranges	all real	$ρ \geq 0$ , $0 \leq ϕ < 2 π$	$r \geq 0$ , $0 \leq θ \leq π$ , $0 \leq ϕ < 2 π$
coordinate surfaces	three planes	cylinder, half-plane, plane	sphere, cone, half-plane
unit vectors	$a_{x}, a_{y}, a_{z}$	$a_{ρ}, a_{ϕ}, a_{z}$	$a_{r}, a_{θ}, a_{ϕ}$
do they vary?	no	$a_{ρ}, a_{ϕ}$ vary with $ϕ$	all three vary with $θ$ and $ϕ$
position vector	$x a_{x} + y a_{y} + z a_{z}$	$ρ a_{ρ} + z a_{z}$	$r a_{r}$
scale factors $h_{1}, h_{2}, h_{3}$	$1, 1, 1$	$1, ρ, 1$	$1, r, r sin θ$

	Cartesian	cylindrical	spherical
$d ℓ$	$d x a_{x} + d y a_{y} + d z a_{z}$	$d ρ a_{ρ} + ρ d ϕ a_{ϕ} + d z a_{z}$	$d r a_{r} + r d θ a_{θ} + r sin θ d ϕ a_{ϕ}$
$d S_{1}, d S_{2}, d S_{3}$	$d y d z$ , $d x d z$ , $d x d y$	$ρ d ϕ d z$ , $d ρ d z$ , $ρ d ρ d ϕ$	$r^{2} sin θ d θ d ϕ$ , $r sin θ d r d ϕ$ , $r d r d θ$
$d V$	$d x d y d z$	$ρ d ρ d ϕ d z$	$r^{2} sin θ d r d θ d ϕ$

Differential elements in any orthogonal system

d ℓ d V = h_{1} d u_{1} a_{1} + h_{2} d u_{2} a_{2} + h_{3} d u_{3} a_{3}, = h_{1} h_{2} h_{3} d u_{1} d u_{2} d u_{3} . (2.3.42)

The scale factor $h_{i}$ is what turns a change in the parameter $u_{i}$ into a change in length. A coordinate that is itself a length ( $x$ , $y$ , $z$ , $ρ$ , $r$ ) has $h = 1$ , whereas an angle has a scale factor equal to the radius of the circle along which that angle moves the point ( $ρ$ for the cylindrical $ϕ$ , $r$ for $θ$ , $r sin θ$ for the spherical $ϕ$ ). The surface element on a coordinate surface $u_{i} = const$ is the product of the other two edges, $d S_{1} = h_{2} h_{3} d u_{2} d u_{3}$ , $d S_{2} = h_{1} h_{3} d u_{1} d u_{3}$ , $d S_{3} = h_{1} h_{2} d u_{1} d u_{2}$ .

In plain words

Learn the three rows "scale factors" of Table 2.3.4 by heart and understand where they come from (lengths have $h = 1$ , angles have $h =$ radius of their circle). Everything else in the table can be reconstructed from them, and so can all four differential operators of Section 2.3.10.

2.3.7 The three coordinate systems at a glance