| ### Statement | | ### Statement |
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| $12.2.1.$ | | $12.2.1.$ |
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| According to the Huygens-Fresnel principle, each section of the wave front is | | 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 | | 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- | | 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 | | 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 | | to the plane of the front; b) the radius of the spherical front increases by τc |
| over time τ. | | over time τ. |
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| ### Solution | | ### Solution |
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| Huygens–Fresnel Principle (recall) | | Huygens–Fresnel Principle (recall) |
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| At time t, the position of the wavefront is known. | | 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). | | 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. | | The new wavefront at time $t+\tau$ is the envelope surface of all those secondary spheres. |
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| Note: c is the speed of light in vacuum. | | Note: c is the speed of light in vacuum. |
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| --- | | --- |
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| Part a) | | Part a) |
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| Initial assumption: | | Initial assumption: |
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| At time t = 0, the wavefront is the plane x = 0 (perpendicular to the x-axis, propagating to the right). | | At time t = 0, the wavefront is the plane x = 0 (perpendicular to the x-axis, propagating to the right). |
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| We apply Huygens–Fresnel: | | We apply Huygens–Fresnel: |
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| · Each point $P(0, y, z)$ on the plane x = 0 emits a secondary spherical wave that reaches a radius $c\tau$ in time $\tau$. | | · 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$. | | · The spheres have centers on the plane x = 0 and all have the same radius $ R = c\tau$. |
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| Geometry of the envelopes: | | 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 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. |
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| The spheres centered at (0, y, z) just touch the plane $ x = c\tau$. | | The spheres centered at (0, y, z) just touch the plane $ x = c\tau$. |
| No sphere reaches beyond $x = c\tau$. | | 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)$. | | 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)$. |
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| Result: | | Result: |
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| 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. | | 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. |
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| Speed: | | Speed: |
| We apply Huygens–Fresnel: | | We apply Huygens–Fresnel: |
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| Each point P on the spherical surface of radius R emits a secondary spherical wave of radius$ c\tau$ after time$ \tau$. | | Each point P on the spherical surface of radius R emits a secondary spherical wave of radius$ c\tau$ after time$ \tau$. |
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| Geometry of the envelope: | | Geometry of the envelope: |
| The centers of the secondary spheres lie on the original sphere of radius R. | | The centers of the secondary spheres lie on the original sphere of radius R. |
| All secondary spheres have the same radius $c\tau$. | | All secondary spheres have the same radius $c\tau$. |
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| The envelope of all these spheres is another sphere concentric with the original one, whose radius is $R + c\tau$. | | The envelope of all these spheres is another sphere concentric with the original one, whose radius is $R + c\tau$. |
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| Justification: | | 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: | | 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: |
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| $OQ = OP + PQ = R + c\tau$ | | $OQ = OP + PQ = R + c\tau$ |
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| Since this holds for all directions, the envelope is the sphere of radius$ R + c\tau$. | | Since this holds for all directions, the envelope is the sphere of radius$ R + c\tau$. |
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| Increase in radius in time $\tau$: | | Increase in radius in time $\tau$: |
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| $\Delta R = (R + c\tau) - R = c\tau$ | | $\Delta R = (R + c\tau) - R = c\tau$ |
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| $\boxed{\text{The radius of a spherical wavefront increases by } c\tau \text{ during a time } \tau.}$ | | $\boxed{\text{The radius of a spherical wavefront increases by } c\tau \text{ during a time } \tau.}$ |
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| #### Answer | | #### Answer |
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| $\boxed{\text{The plane wavefront moves with speed } c \text{ perpendicular to the plane.}}$ | | $\boxed{\text{The plane wavefront moves with speed } c \text{ perpendicular to the plane.}}$ |
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| $\boxed{\text{The radius of a spherical wavefront increases by } c\tau \text{ during a time } \tau.}$ | | $\boxed{\text{The radius of a spherical wavefront increases by } c\tau \text{ during a time } \tau.}$ |
