Q42P - Introduction to Electrodynamics

Question

Express the cylindrical unit vectors $\hat{\mathbf{s}}\mathbf{,}\mathbf{ }\hat{\mathbf{\varphi }}\mathbf{,}\mathbf{ }\hat{\mathbf{z}}$   in terms of $\hat{\mathbf{x}}\mathbf{,}\mathbf{ }\hat{\mathbf{y}}\mathbf{,}\mathbf{ }\hat{\mathbf{z}}$  (that is, derive Eq. 1.75). "Invert" your formulas to get  $\hat{\mathbf{x}}\mathbf{,}\mathbf{ }\hat{\mathbf{y}}\mathbf{,}\mathbf{ }\hat{\mathbf{z}}$ in terms of  $\hat{\mathbf{s}}\mathbf{,}\mathbf{ }\hat{\mathbf{\varphi }}\mathbf{,}\mathbf{ }\hat{\mathbf{z}}$

Step-by-Step Solution

Answer

It is obtained that$\overbrace{x}=\left(\cos \psi \right)\stackrel{⏜}{s}-\left(\sin \psi \right)\stackrel{⏜}{\varphi },\stackrel{⏜}{y}=\left(\sin \psi \right)\stackrel{⏜}{s}+\left(\cos \psi \right)\stackrel{⏜}{s},and \stackrel{⏜}{z}= \stackrel{⏜}{z}.$

Step 1: Define cylindrical coordinates

In cylindrical coordinates, the point   is represented as $P=(s, \varphi , z)$  , where $P=(s, \varphi , z)$  is distance of point P from z axis, the azimuthal angle, and coordinate of point P on z-axis respectively, as shown in following figure:

               

From the figure, write:

$$\begin{gathered} x=s \cos \varphi \\ y=s \sin \varphi \\ z=z \end{gathered}$$

The unit vectors in cylindrical coordinates are:

$$\begin{gathered} \stackrel{⏜}{s}=\left(\cos \varphi \right)\stackrel{⏜}{x}+\left(\sin \varphi \right)\stackrel{⏜}{y} \\ \stackrel{⏜}{\varphi }=-\left(\sin \varphi \right)\stackrel{⏜}{x}+\left(\cos \varphi \right)\stackrel{⏜}{y} \\ \stackrel{⏜}{z}=\stackrel{⏜}{z} \end{gathered}$$

The displacement vector is given as $dl=dx \hat{x}+dy \hat{y}+dz \hat{z}.$ Differentiate transformation equation with respect to s .

$$\begin{gathered} dx=\left(\cos \varphi \right)ds \\ dy=\left(\sin \varphi \right)ds \\ dz=0 \\ \mathrm{The} \mathrm{displacement} \mathrm{vector} \mathrm{now} \mathrm{becomes}: \\ \mathrm{dl}=\left(\mathrm{cos\varphi }\right)\mathrm{ds}\stackrel{⏜}{ \mathrm{x}}+\left(\mathrm{sin\varphi }\right)\mathrm{ds} \stackrel{⏜}{\mathrm{y}}+0\stackrel{⏜}{\mathrm{z}} \\ =\mathrm{ds}\left(\left(\mathrm{cos\varphi }\right)\stackrel{⏜}{\mathrm{x}}+\left(\mathrm{sin\varphi }\right)\stackrel{⏜}{\mathrm{y}}\right) \\ \mathrm{Compare} \mathrm{above} \mathrm{equation} \mathrm{with} \mathrm{dl}=\mathrm{ds}\stackrel{⏜}{ \mathrm{s}},\mathrm{we} \mathrm{get} \\ \stackrel{⏜}{\mathrm{s}}=\left(\mathrm{cos\varphi }\right)\stackrel{⏜}{\mathrm{y}}+\left(\mathrm{sin\varphi }\right)\stackrel{⏜}{\mathrm{y}} \end{gathered}$$

Step: 2 Compute unit vector s ^ .

The displacement vector is given as$d{l}_{\varphi }=dx \hat{x},dy \hat{y},dz \hat{z}$  . Differentiate transformation equation with respect to $\varphi$ .

 $$\begin{gathered} dx=s \sin \varphi d\varphi \\ dy=s \cos \varphi d\varphi \\ dz=0 \end{gathered}$$

The displacement vector now becomes:

 $$\begin{gathered} dl=s \sin \varphi d\varphi \hat{x} +s\cos \varphi d\varphi \hat{y}+0 \hat{z} \\ =ds\left((s \sin \varphi d\varphi )\hat{x} +\left(s\cos \varphi d\varphi \right) \hat{y}\right) \end{gathered}$$

Compare above equation with $dl=sd\varphi \hat{\varphi }$  , we get,

 $\hat{s}=(-\sin \varphi )\hat{x}+\left(\cos \varphi \right)\hat{y}$

Step: 3 Compute unit vector x ^

$$\begin{gathered} As \stackrel{⏜}{s}=\left(\cos \varphi \right)\stackrel{⏜}{x}+\left(\sin \varphi \right)\stackrel{⏜}{y},\mathrm{multiple} \mathrm{cos\phi } \mathrm{on} \mathrm{both} \mathrm{sides} \mathrm{of} \\ \stackrel{⏜}{\mathrm{s}}=\left(\mathrm{cos\varphi }\right)\stackrel{⏜}{\mathrm{x}}+\left(\mathrm{sin\varphi }\right)\stackrel{⏜}{\mathrm{y}} \mathrm{as}, \\ \left(\mathrm{cos\phi }\right)\stackrel{⏜}{\mathrm{s}}=\left({\cos }^2\mathrm{\varphi }\right)\stackrel{⏜}{\mathrm{x}}+\left(\mathrm{sin\varphi cos\varphi }\right)\stackrel{⏜}{\mathrm{y}} \\ \mathrm{Now}, \mathrm{multiply} \sin \varphi \mathrm{on} \mathrm{both} \mathrm{sides} \mathrm{of} \stackrel{⏜}{\varphi }=-\left(\mathrm{sin\varphi }\right)\stackrel{⏜}{\mathrm{x}}+\left(\mathrm{cos\varphi }\right)\stackrel{⏜}{\mathrm{y} }\mathrm{as}, \\ \left(\mathrm{sin\phi }\right)\stackrel{⏜}{\mathrm{\varphi }}=\left(-{\sin }^2\mathrm{\varphi }\right)\stackrel{⏜}{\mathrm{x}}+\left(\sin \varphi c\mathrm{os}\varphi \right)\stackrel{⏜}{\mathrm{y} } \end{gathered}$$

$$\begin{gathered} \mathrm{subtract} \left(\sin \phi \right)\stackrel{⏜}{\varphi }=\left(-{\sin }^2\varphi \right)\stackrel{⏜}{x}+\left(\sin \varphi \cos \varphi \right)\stackrel{⏜}{y} \mathrm{from} \\ \left(\mathrm{cos\phi }\right)\stackrel{⏜}{\mathrm{s}}=\left(\mathrm{cos\varphi }\right)\stackrel{⏜}{\mathrm{x}}+\left(\mathrm{sin\varphi cos\varphi }\right)\stackrel{⏜}{\mathrm{y}} \mathrm{as}, \\ \left(\mathrm{cos\phi }\right)\stackrel{⏜}{\mathrm{s}}=-\left(\mathrm{sin\phi }\right)\stackrel{⏜}{\mathrm{\varphi }}=\left({\cos }^2\mathrm{\varphi }+{\sin }^2\mathrm{\varphi }\right)\stackrel{⏜}{\mathrm{x}} \\ \stackrel{⏜}{\mathrm{x}}=\left(\mathrm{cos\phi }\right)\stackrel{⏜}{\mathrm{s}}-\left(\mathrm{sin\phi }\right)\stackrel{⏜}{\mathrm{\varphi }} \end{gathered}$$

Step: 4 Compute unit vector y ^

$$\begin{gathered} \mathrm{As} \stackrel{⏜}{\mathrm{s}}=\left(\mathrm{cos\varphi }\right)\stackrel{⏜}{\mathrm{x}}+\left(\mathrm{sin\varphi }\right)\stackrel{⏜}{\mathrm{y}},\mathrm{multiply} \sin \varphi \mathrm{on} \mathrm{both} \mathrm{sides} \mathrm{of} \\ \stackrel{⏜}{\mathrm{s}}=\left(\mathrm{cos\varphi }\right)\stackrel{⏜}{\mathrm{x}}+\left(\mathrm{sin\varphi }\right)\stackrel{⏜}{\mathrm{y}} \mathrm{as}, \\ \left(\sin \varphi \right)\stackrel{⏜}{\mathrm{x}}=\left(\sin \varphi \cos \varphi \right)\stackrel{⏜}{\mathrm{x}}+\left({\sin }^2\varphi \right)\stackrel{⏜}{\mathrm{y}} \\ \mathrm{Now}, \mathrm{multiply} \cos \varphi \mathrm{on} \mathrm{both} \mathrm{sides} \mathrm{of} \stackrel{⏜}{\mathrm{\varphi }}=\left(-\sin \varphi \right)\stackrel{⏜}{\mathrm{x}}+\left(\cos \varphi \right)\stackrel{⏜}{\mathrm{y}} \mathrm{as}, \\ \left(\mathrm{cos\varphi }\right)\stackrel{⏜}{\mathrm{\varphi }}=-\left(-sin\varphi cos\varphi \right)\stackrel{⏜}{\mathrm{x}}+\left(\cos { }^2\varphi \right)\stackrel{⏜}{\mathrm{y}} \\ \mathrm{Add} \mathrm{equations} \left(\mathrm{cos\varphi }\right)\stackrel{⏜}{\mathrm{\varphi }}=\left(-\mathrm{sin\varphi cos\varphi }\right)\stackrel{⏜}{\mathrm{x}}+\left(\cos { }^2\mathrm{\varphi }\right)\stackrel{⏜}{\mathrm{y}} \mathrm{and} \\ \left(\mathrm{sin\varphi }\right)\stackrel{⏜}{\mathrm{x}}=\left(\mathrm{sin\varphi cos\varphi }\right)\stackrel{⏜}{\mathrm{x}}+\left({\sin }^2\mathrm{\varphi }\right)\stackrel{⏜}{\mathrm{y}} \mathrm{as}, \\ \left(\mathrm{sin\phi }\right)\stackrel{⏜}{\mathrm{x}}+\left(\mathrm{cos\phi }\right)\stackrel{⏜}{\mathrm{\varphi }}=\left({\cos }^2\mathrm{\varphi }+{\sin }^2\mathrm{\varphi }\right)\stackrel{⏜}{\mathrm{y}} \\ \stackrel{⏜}{\mathrm{y}}=\left(\mathrm{sin\phi }\right)\stackrel{⏜}{\mathrm{s}}+\left(\mathrm{cos\phi }\right)\stackrel{⏜}{\mathrm{\varphi }} \\ \mathrm{Therefore}, \mathrm{the} \mathrm{required} \mathrm{equations} \mathrm{are} \stackrel{⏜}{\mathrm{x}}=\left(\mathrm{cos\phi }\right)\stackrel{⏜}{\mathrm{s}}-\left(\mathrm{sin\phi }\right)\stackrel{⏜}{\mathrm{x}}; \\ \stackrel{⏜}{\mathrm{y}}=\left(\mathrm{sin\phi }\right)\stackrel{⏜}{\mathrm{s}}+\left(\mathrm{cos\phi }\right)\stackrel{⏜}{\mathrm{\varphi }} \mathrm{and} \stackrel{⏜}{\mathrm{z}}=\stackrel{⏜}{\mathrm{z}}. \end{gathered}$$