Question

What is the meaning of wavefront? Explain reflection of light by the Huygen's wave theory.

Hint:

When drawing the diagram for Huygen's proof of reflection or refraction, the key is to correctly identify the distances traveled by the wavelets in the given time. For reflection, the speed is the same, so \(BC = AD\). For refraction, the speeds are different, so the radii of the wavelets will be different.

Answer 1

Meaning of Wavefront:
A wavefront is defined as the locus of all points in a medium that are vibrating in the same phase. For example, if you drop a stone in a pond, the ripples spread out in circles; each circle represents a wavefront where all particles are at the same stage of their oscillation. \begin{itemize} \item For a point source, the wavefronts are spherical. \item For a line source, the wavefronts are cylindrical. \item At a large distance from a source, a small portion of a spherical or cylindrical wavefront can be considered a plane wavefront. \end{itemize} The direction of wave propagation is always perpendicular to the wavefront.
Explanation of Reflection using Huygen's Theory:
Huygen's principle states that every point on a wavefront acts as a source of secondary spherical wavelets. The new wavefront at any later time is the forward envelope (tangent) of these secondary wavelets. Let's use this to prove the law of reflection (\(i=r\)). Explanation: \begin{enumerate} \item Let XY be a reflecting surface and AB be an incident plane wavefront making an angle of incidence \(i\) with the surface. \item At time \(t=0\), the wavefront touches the surface at point A. According to Huygen's principle, A becomes a source of secondary wavelets. \item The point B on the wavefront will travel a distance BC to reach the surface at point C. Let the time taken be \(t\). Then, \(BC = vt\), where \(v\) is the speed of light. \item In the same time \(t\), the secondary wavelet from A will spread out in the same medium as a hemisphere of radius \(AD = vt\). \item The reflected wavefront is the common tangent CD drawn from point C to the wavelet originating from A. The angle of reflection is \(r\). \end{enumerate} Proof of Law of Reflection: Consider the two right-angled triangles, \(\triangle ABC\) and \(\triangle ADC\). \begin{itemize} \item The side AC is common to both triangles. \item The side \(BC = vt\) and the radius of the wavelet \(AD = vt\). Thus, \(BC = AD\). \item Both are right-angled triangles (\(\angle B = \angle D = 90^\circ\)). \end{itemize} By the Right-angle-Hypotenuse-Side (RHS) congruence criterion, \(\triangle ABC \cong \triangle ADC\). Therefore, the corresponding angles must be equal. \[ \angle BAC = \angle DCA \] From the geometry of the diagram, \(\angle BAC = i\) (angle of incidence) and \(\angle DCA = r\) (angle of reflection). \[ i = r \] This proves the law of reflection.