Consider each of the electric- and magnetic-field orientations given next. In each case, what is the direction of propagation of the wave? (a)
in the x
-direction, b
in the y
- direction; (b)
in the y - direction x
in the b
-direction;
in the z
-direction,
in the y
-direction; (d)
in the e
-direction,
in the z
-direction.
The Correct Answer and Explanation is:
To determine the direction of propagation of the electromagnetic wave, we can use the right-hand rule. The direction of propagation is given by the vector cross product of the electric field E and the magnetic field B. Specifically:
- Direction of propagation: v=E×B\mathbf{v} = \mathbf{E} \times \mathbf{B}v=E×B
Now, let’s analyze each case one by one.
(a) E\mathbf{E}E in the x-direction, B\mathbf{B}B in the y-direction
- Given that the electric field is along the x-axis, and the magnetic field is along the y-axis, we can find the direction of the wave propagation by calculating the cross product: v=E×B\mathbf{v} = \mathbf{E} \times \mathbf{B}v=E×B According to the right-hand rule, if you point your fingers in the direction of E (x-axis) and curl them toward B (y-axis), your thumb will point in the direction of v (the direction of wave propagation).
The resulting direction of propagation will be along the z-axis.
Thus, the wave propagates in the z-direction.
(b) E\mathbf{E}E in the z-direction, B\mathbf{B}B in the x-direction
- Here, the electric field is along the z-axis, and the magnetic field is along the x-axis. Using the right-hand rule again: v=E×B\mathbf{v} = \mathbf{E} \times \mathbf{B}v=E×B Point your fingers in the direction of E (z-axis) and curl them toward B (x-axis), your thumb will point in the y-direction.
Thus, the wave propagates in the y-direction.
(c) E\mathbf{E}E in the z-direction, B\mathbf{B}B in the y-direction
- In this case, the electric field is along the z-axis, and the magnetic field is along the y-axis: v=E×B\mathbf{v} = \mathbf{E} \times \mathbf{B}v=E×B Point your fingers in the direction of E (z-axis) and curl them toward B (y-axis), your thumb will point in the x-direction.
Thus, the wave propagates in the x-direction.
(d) E\mathbf{E}E in the y-direction, B\mathbf{B}B in the z-direction
- Finally, for the electric field along the y-axis and the magnetic field along the z-axis: v=E×B\mathbf{v} = \mathbf{E} \times \mathbf{B}v=E×B Point your fingers in the direction of E (y-axis) and curl them toward B (z-axis), your thumb will point in the x-direction.
Thus, the wave propagates in the x-direction.
Summary:
- (a) Propagation in the z-direction
- (b) Propagation in the y-direction
- (c) Propagation in the x-direction
- (d) Propagation in the x-direction
This relationship between the electric and magnetic fields and the wave propagation direction is a fundamental principle of electromagnetic waves, governed by Maxwell’s equations.
