Interpreting ##\hat{e}_z## in Maxwell's equations

• flintbox
In summary, the conversation is about trying to interpret a form of Maxwell's equations, specifically an equation involving the term $\^{e}_z$. The questioner is asking for clarification on the notation and the meaning of the equation, which is assumed to be for a plane wave field. The responder explains that the term comes from taking the curl of the B field, but notes that the equation is not entirely correct.

flintbox

Hi, I'm trying to interpret a form of Maxwell's equations, but I can't seem to figure out where the term $\^{e}_z$ comes from in the following equation:
##
\frac{\partial{\vec{E}_t}}{\partial{z}}+i\frac{\omega}{c}\hat{e}_z\times \vec{B}_t=\vec{\nabla}_tE_z
##

This is a guess because more background on the source of this equation is needed. My guess is that the problem being worked is assuming a plane wave field dependence?

Will you please clarify the notations? What does ##t## stand for?

flintbox said:
Hi, I'm trying to interpret a form of Maxwell's equations, but I can't seem to figure out where the term $\^{e}_z$ comes from in the following equation:
##
\frac{\partial{\vec{E}_t}}{\partial{z}}+i\frac{\omega}{c}\hat{e}_z\times \vec{B}_t=\vec{\nabla}_tE_z
##
It comes from taking the curl of the B field (here they are assuming a plane wave). But the equation is not completely correct, the term on the rhs does not make any sense.

1. What is ##\hat{e}_z## in Maxwell's equations?

##\hat{e}_z## is a unit vector in the z-direction, also known as the "z-hat" vector. It is commonly used in vector calculus to represent a unit vector in three-dimensional space.

2. Why is ##\hat{e}_z## important in Maxwell's equations?

In Maxwell's equations, ##\hat{e}_z## is used to represent the direction of the electric field in the z-direction. This is important because Maxwell's equations describe the behavior of electromagnetic fields, and the z-direction is a crucial component of these fields.

3. How is ##\hat{e}_z## related to other unit vectors in Maxwell's equations?

##\hat{e}_z## is just one of three unit vectors used in Maxwell's equations. The other two are ##\hat{e}_x## and ##\hat{e}_y##, which represent the x and y directions, respectively. Together, these three unit vectors form a basis for describing three-dimensional space.

4. Can ##\hat{e}_z## have a different direction in different coordinate systems?

Yes, the direction of ##\hat{e}_z## can vary depending on the coordinate system being used. In Cartesian coordinates, it always points in the z-direction, but in other coordinate systems, such as spherical or cylindrical coordinates, its direction may change.

5. How can I visualize ##\hat{e}_z## in Maxwell's equations?

One way to visualize ##\hat{e}_z## is to imagine a unit vector pointing straight up from the origin of a three-dimensional coordinate system. This vector represents the direction of ##\hat{e}_z## and can be used to visualize the z-direction in Maxwell's equations.