Refraction And Reflection Of Plane Waves Using Huygens Principle

Using Snell's law: where (air), , : Thus, .

The total current in the circuit is: where is the internal resistance. The terminal voltage is given by: Rearranging:

Huygens' principle states that every point on a wavefront is a source of secondary wavelets that spread out in the forward direction at the same speed as the original wave. The new wavefront at any time is the envelope of all these secondary wavelets.
For the laws of refraction:
- The incident ray, the refracted ray, and the normal to the surface at the point of incidence all lie in the same plane.
- The ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant and is equal to the refractive index of the medium, i.e., .

Huygens' Principle states that every point on a wavefront can be considered as a source of secondary wavelets, and the position of the wavefront at any later time is the surface tangent to these secondary wavelets. In other words, each point on the wavefront acts as a secondary source of waves, and the new wavefront is formed by the envelope of all these secondary waves.
This principle can be used to explain the laws of reflection of light:
1. Law of Reflection (First Law):
The angle of incidence ( ) is equal to the angle of reflection ( ), i.e., This is confirmed by Huygens' principle because, when the wavefront strikes a reflective surface, each point on the wavefront produces secondary wavelets. These wavelets propagate and form a new wavefront that follows the law of reflection.
2. Law of Reflection (Second Law):
The incident ray, the reflected ray, and the normal to the surface of reflection all lie in the same plane. This can be understood by imagining the wavelets from the incident wavefront interacting with the surface, forming a new wavefront that obeys this geometrical constraint.
Thus, Huygens' principle provides a clear explanation for the reflection of light, where the wavefront interacts with the reflecting surface, and the reflected wavefront is formed by the secondary wavelets.

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 ( ). Explanation: \begin{enumerate} \item Let XY be a reflecting surface and AB be an incident plane wavefront making an angle of incidence with the surface. \item At time , 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 . Then, , where is the speed of light. \item In the same time , the secondary wavelet from A will spread out in the same medium as a hemisphere of radius . \item The reflected wavefront is the common tangent CD drawn from point C to the wavelet originating from A. The angle of reflection is . \end{enumerate} Proof of Law of Reflection: Consider the two right-angled triangles, and . \begin{itemize} \item The side AC is common to both triangles. \item The side and the radius of the wavelet . Thus, . \item Both are right-angled triangles ( ). \end{itemize} By the Right-angle-Hypotenuse-Side (RHS) congruence criterion, . Therefore, the corresponding angles must be equal. From the geometry of the diagram, (angle of incidence) and (angle of reflection). This proves the law of reflection.