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.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:
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\frac{\partial{\vec{E}_t}}{\partial{z}}+i\frac{\omega}{c}\hat{e}_z\times \vec{B}_t=\vec{\nabla}_tE_z
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##\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.
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.
##\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.
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.
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.