# Writing an expression for the scalar value of the electric field $E$

I am currently studying Optics, fifth edition, by Hecht. In chapter 2.7 Plane Waves, the following example is given:

An electromagnetic plane wave is described by its electric field $$E$$. The wave has an amplitude $$E_0$$, an angular frequency $$\omega$$, a wavelength $$\lambda$$, and travels at speed $$c$$ outward in the direction of the unit propagation vector

$$\hat{\mathbf{k}} = (4\hat{\mathbf{i}} + 2\hat{\mathbf{j}})/\sqrt{20}$$

(not to be confused with the unit basis vector $$\hat{\mathbf{k}}$$). Write an expression for the scalar value of the electric field $$E$$.

Solution

We want an equation of the form

$$E(x, y, z, t) = E_0 e^{i\hat{\mathbf{k}} \cdot (\vec{\mathbf{r}} - \omega t)}$$

Here

$$\vec{\mathbf{k}} \cdot \vec{\mathbf{r}} = \dfrac{2\pi}{\lambda} \hat{\mathbf{k}} \cdot \vec{\mathbf{r}}$$

and

$$\vec{\mathbf{k}} \cdot \vec{\mathbf{r}} = \dfrac{2 \pi}{\lambda \sqrt{20}}(4\hat{\mathbf{i}} + 2\hat{\mathbf{j}}) \cdot (x\hat{\mathbf{i}} + y\hat{\mathbf{j}} + z\hat{\mathbf{k}})$$

$$\vec{\mathbf{k}} \cdot \vec{\mathbf{r}} = \dfrac{\pi}{\lambda \sqrt{5}}(4x + 2y)$$

Hence

$$E = E_0e^{i \left[ \dfrac{\pi}{\lambda \sqrt{5}}(4x + 2y) - \omega t \right]}$$

I have the following questions:

1. Shouldn't $$E(x, y, z, t) = E_0 e^{i\hat{\mathbf{k}} \cdot (\vec{\mathbf{r}} - \omega t)}$$ be $$E(x, y, z, t) = E_0 e^{i(\hat{\mathbf{k}} \cdot \vec{\mathbf{r}} - \omega t)}$$?

2. Where did $$\vec{\mathbf{k}} \cdot \vec{\mathbf{r}} = \dfrac{2\pi}{\lambda} \hat{\mathbf{k}} \cdot \vec{\mathbf{r}}$$ come from? What is the reasoning for this?

I would greatly appreciate it if people would please take the time to clarify these points.

1. Shouldn't $$E(x, y, z, t) = E_0 e^{i\hat{\mathbf{k}} \cdot (\vec{\mathbf{r}} - \omega t)}$$ be $$E(x, y, z, t) = E_0 > e^{i(\hat{\mathbf{k}} \cdot \vec{\mathbf{r}} - \omega t)}$$?
1. Where did $$\vec{\mathbf{k}} \cdot \vec{\mathbf{r}} = \dfrac{2\pi}{\lambda} \hat{\mathbf{k}} \cdot \vec{\mathbf{r}}$$ come from? What is the reasoning for this?
The hat indicates a unit vector. Hence $$\hat k = \vec k /|\vec k|$$. as you probably know the wavenumber (=magnitude of the vector $$\vec k$$) is given by $$k=2\pi/\lambda$$