Electromagnetic waves > Propagation of electromagnetic waves > Problem 12.2.1

Statement

$12.2.1.$

According to the Huygens-Fresnel principle, each section of the wave front is
a source of a secondary spherical wave. The envelope of these waves gives a
new wave front. Show, using this principle, that: a) the plane front of an elec-
tromagnetic wave moves at the speed of light c in the direction perpendicular
to the plane of the front; b) the radius of the spherical front increases by τc
over time τ.

Solution

Huygens–Fresnel Principle (recall)

At time t, the position of the wavefront is known.
Each point on that wavefront becomes a source of secondary spherical waves of radius c\tau (where \tau is the considered time interval).
The new wavefront at time $t+\tau$ is the envelope surface of all those secondary spheres.

Note: c is the speed of light in vacuum.

Part a)

Initial assumption:

At time t = 0, the wavefront is the plane x = 0 (perpendicular to the x-axis, propagating to the right).

We apply Huygens–Fresnel:

· Each point $P(0, y, z)$ on the plane x = 0 emits a secondary spherical wave that reaches a radius $c\tau$ in time $\tau$.
· The spheres have centers on the plane x = 0 and all have the same radius $ R = c\tau$.

Geometry of the envelopes:
The envelope of a set of spheres of equal radius whose centers lie on a plane is another parallel plane, located at a distance equal to the radius.

The spheres centered at (0, y, z) just touch the plane $ x = c\tau$.
No sphere reaches beyond $x = c\tau$.
The plane $x = c\tau$ is tangent to all spheres (because the minimum distance from any center (0, y, z) to the plane $x = c\tau$ is exactly f c\tau)$.

Result:

The new wavefront at time $\tau $is the plane$ x = c\tau$. Therefore, in time $\tau$ the wavefront has advanced a distance $c\tau$ in the direction perpendicular to itself.

Speed:

$v = \frac{\text{displacement}}{\text{time}} = \frac{c\tau}{\tau} = c$

$\boxed{\text{The plane wavefrontmoves with speed } c \text{ perpendicular to the plane.}}$

Part b)

We apply Huygens–Fresnel:

Each point P on the spherical surface of radius R emits a secondary spherical wave of radius$ c\tau$ after time$ \tau$.

Geometry of the envelope:
The centers of the secondary spheres lie on the original sphere of radius R.
All secondary spheres have the same radius $c\tau$.

The envelope of all these spheres is another sphere concentric with the original one, whose radius is $R + c\tau$.

Justification:
A point Q is on the outer envelope if there exists a center P on the original sphere such that Q lies on the corresponding secondary sphere and, moreover, the direction $O \to P \to Q$ is radial. In that case:

$OQ = OP + PQ = R + c\tau$

Since this holds for all directions, the envelope is the sphere of radius$ R + c\tau$.

Increase in radius in time $\tau$:

$\Delta R = (R + c\tau) - R = c\tau$

$\boxed{\text{The radius of a spherical wavefront increases by } c\tau \text{ during a time } \tau.}$

Answer

$\boxed{\text{The plane wavefront moves with speed } c \text{ perpendicular to the plane.}}$

$\boxed{\text{The radius of a spherical wavefront increases by } c\tau \text{ during a time } \tau.}$