Comprehensive Study Guide on Geometric Optics: Reflection and Spherical Mirrors

Fundamentals of Reflection and the Law of Reflection

• Definition of Reflection: Reflection is the change in direction of a wavefront at an interface between two different media so that the wavefront returns into the medium from which it originated.
• Key Terminology:
  • Incident Ray: The incoming ray of light approaching a reflective surface.
  • Reflected Ray: The ray of light that bounces off the reflective surface.
  • Normal Line: An imaginary perpendicular line drawn to the reflective surface at the point of incidence ($90^\times$ to the surface).
  • Angle of Incidence ($\theta_i$): The angle measured between the incident ray and the normal line.
  • Angle of Reflection ($\theta_r$): The angle measured between the reflected ray and the normal line.
• The Law of Reflection:
  1. The incident ray, the reflected ray, and the normal to the surface at the point of incidence all lie within the exact same geometric plane.
  2. The angle of incidence is equal to the angle of reflection:         $\theta_i = \theta_r$

Analysis of Plane Mirror Reflections: Single Surface Calculations

• Scenario Parameters: A light ray strikes a flat plane reflective surface at an angle of $56^\times$ relative to the surface itself.
• Geometric Setup:
  • Angle between ray and surface: $\alpha = 56^\times$
  • Angle between surface and normal line: $90^\times$
• Part (a) Calculation of the Angle of Incidence ($\theta_i$):
  • The angle of incidence is measured with respect to the normal line, not the surface.
  • $\theta_i = 90^\times - \alpha$
  • $\theta_i = 90^\times - 56^\times = 34^\times$
• Part (b) Calculation of the Angle of Reflection ($\theta_r$):
  • By applying the Law of Reflection ($\theta_r = \theta_i$):
  • $\theta_r = 34^\times$
• Part (c) Calculation of the Angle Made by the Reflected Ray and the Surface ($\beta$):
  • The angle between the reflected ray and the plane surface is complementary to the angle of reflection:
  • $\beta = 90^\times - \theta_r$
  • $\beta = 90^\times - 34^\times = 56^\times$
• Part (d) Calculation of the Total Angle Made by the Incident and Reflected Rays ($\phi$):
  • The angular separation between the incoming incident ray and outgoing reflected ray is the sum of both angles:
  • $\phi = \theta_i + \theta_r$
  • $\phi = 34^\times + 34^\times = 68^\times$

Reflection Across Multiple Intersecting Plane Mirrors

• System Setup:
  • Two flat plane mirrors, designated as $M_1$ and $M_2$, are inclined toward each other such that their surfaces meet at an interior vertex angle of $\Phi = 120^\times$.
  • An incident ray strikes mirror $M_1$ at an angle of incidence $\theta_{i1} = 65^\times$.
• Step 1: First Reflection at Mirror $M_1$:
  • Angle of incidence at $M_1$: $\theta_{i1} = 65^\times$
  • By the Law of Reflection, the angle of reflection at $M_1$ is $\theta_{r1} = 65^\times$.
  • The angle formed between the reflected ray leaving $M_1$ and the surface of mirror $M_1$ ($\alpha_1$) is:         $\alpha_1 = 90^\times - 65^\times = 25^\times$
• Step 2: Triangle Geometry Between Mirrors:
  • The path of the light ray moving from mirror $M_1$ to mirror $M_2$ forms an interior triangle bounded by mirror $M_1$, mirror $M_2$, and the ray itself.
  • The three interior angles of this triangle must sum to $180^\times$:         $\Phi + \alpha_1 + \alpha_2 = 180^\times$
  • Substitute known values ($\Phi = 120^\times$, $\alpha_1 = 25^\times$):         $120^\times + 25^\times + \alpha_2 = 180^\times$$145^\times + \alpha_2 = 180^\times$$\alpha_2 = 180^\times - 145^\times = 35^\times$
  • $\alpha_2 = 35^\times$ represents the angle between the surface of mirror $M_2$ and the ray hitting mirror $M_2$.
• Step 3: Second Reflection at Mirror $M_2$:
  • The angle of incidence on mirror $M_2$ ($\theta_{i2}$) is calculated relative to the normal of mirror $M_2$:         $\theta_{i2} = 90^\times - \alpha_2$$\theta_{i2} = 90^\times - 35^\times = 55^\times$
  • By the Law of Reflection, the final exit angle $\theta$ at which the ray leaves mirror $M_2$ relative to its normal is:         $\theta = \theta_{r2} = \theta_{i2} = 55^\times$

Theoretical Derivation of the Spherical Mirror Equation

• Geometric Parameters:
  • $O$ = Object location on the principal axis
  • $I$ = Image location on the principal axis
  • $V$ = Vertex (pole) of the spherical mirror surface
  • $C$ = Center of curvature of the mirror surface
  • $F$ = Focal point of the mirror
  • $R$ = Radius of curvature (distance from $V$ to $C$)
  • $f$ = Focal length (distance from $V$ to $F$)
  • $d_o$ = Object distance from vertex $V$
  • $d_i$ = Image distance from vertex $V$
  • $h_o$ = Object height
  • $h_i$ = Image height
• Derivation Steps using Similar Triangles:
  1. Consider a ray passing through object top $O$ striking the vertex $V$ at angle $\theta$ to the principal axis. It reflects at angle $\theta$ through image top $I$.
  2. The right-angled triangle formed by object height $h_o$ and object distance $d_o$ gives:         $\tan(\theta) = \frac{h_o}{d_o}$
  3. The right-angled triangle formed by inverted image height $-h_i$ and image distance $d_i$ gives:         $\tan(\theta) = \frac{-h_i}{d_i}$
  4. Equating the tangent expressions defines the lateral magnification ($m$):         $m = \frac{h_i}{h_o} = -\frac{d_i}{d_o}$
  5. Now consider a ray passing through the center of curvature $C$. The triangles formed by object and image relative to point $C$ are similar:         $\frac{h_o}{-h_i} = \frac{d_o - R}{R - d_i}$
  6. Substitute the relation $\frac{h_o}{-h_i} = \frac{d_o}{d_i}$ into the equation:         $\frac{d_o}{d_i} = \frac{d_o - R}{R - d_i}$
  7. Cross-multiply and expand the terms:         $d_o(R - d_i) = d_i(d_o - R)$$d_o R - d_o d_i = d_i d_o - d_i R$$d_o R + d_i R = 2 d_o d_i$
  8. Divide all terms by $(d_o d_i R)$:$\frac{d_o R}{d_o d_i R} + \frac{d_i R}{d_o d_i R} = \frac{2 d_o d_i}{d_o d_i R}$$\frac{1}{d_i} + \frac{1}{d_o} = \frac{2}{R}$
  9. For small-angle paraxial approximations, the focal length is exactly half of the radius of curvature ($f = \frac{R}{2}$ or $\frac{2}{R} = \frac{1}{f}$).
  10. Substituting $f$ into the equation yields the universal Mirror Equation:         $\frac{1}{d_o} + \frac{1}{d_i} = \frac{1}{f}$

Quantitative Image Location and Characterization for Spherical Mirrors

• System Specification: A concave spherical mirror has a focal length $f = +10.0\,\text{cm}$. (A positive focal length indicates a concave converging mirror).

• Case 1: Object Distance $d_o = 25.0\,\text{cm}$

  • Calculation of Image Distance ($d_i$):$\frac{1}{d_o} + \frac{1}{d_i} = \frac{1}{f}$$\frac{1}{25.0} + \frac{1}{d_i} = \frac{1}{10.0}$$\frac{1}{d_i} = \frac{1}{10.0} - \frac{1}{25.0}$$\frac{1}{d_i} = \frac{5}{50.0} - \frac{2}{50.0} = \frac{3}{50.0}\,\text{cm}^{-1}$$d_i = \frac{50.0}{3}\,\text{cm} \approx +16.67\,\text{cm}$
  • Calculation of Magnification ($m$):$m = -\frac{d_i}{d_o} = -\frac{16.67}{25.0} = -0.667$
  • Image Description:
    • Location: Formed at $d_i = +16.67\,\text{cm}$ in front of the mirror surface (between focal point $F$ and center of curvature $C$).
    • Real vs. Virtual: Real image (positive $d_i$ value; rays physically converge).
    • Orientation: Inverted (negative magnification $m = -0.667$).
    • Size: Diminished / Reduced ($|m| < 1$; image is $66.7\%$ of original object size).

• Case 2: Object Distance $d_o = 10.0\,\text{cm}$

  • Calculation of Image Distance ($d_i$):$\frac{1}{d_o} + \frac{1}{d_i} = \frac{1}{f}$$\frac{1}{10.0} + \frac{1}{d_i} = \frac{1}{10.0}$$\frac{1}{d_i} = \frac{1}{10.0} - \frac{1}{10.0} = 0$$d_i = \frac{1}{0} \rightarrow \infty$
  • Image Description:
    • Location: Formed at infinity ($d_i = \infty$).
    • Ray Behavior: Reflected rays travel completely parallel to one another and never intersect.
    • Image State: No focused image is formed at any finite position.

• Case 3: Object Distance $d_o = 5.00\,\text{cm}$

  • Calculation of Image Distance ($d_i$):$\frac{1}{d_o} + \frac{1}{d_i} = \frac{1}{f}$$\frac{1}{5.00} + \frac{1}{d_i} = \frac{1}{10.0}$$\frac{1}{d_i} = \frac{1}{10.0} - \frac{1}{5.00}$$\frac{1}{d_i} = \frac{1}{10.0} - \frac{2}{10.0} = -\frac{1}{10.0}\,\text{cm}^{-1}$$d_i = -10.0\,\text{cm}$
  • Calculation of Magnification ($m$):$m = -\frac{d_i}{d_o} = -\frac{-10.0}{5.00} = +2.00$
  • Image Description:
    • Location: Formed at $d_i = -10.0\,\text{cm}$ behind the mirror surface.
    • Real vs. Virtual: Virtual image (negative $d_i$ value; rays diverge and must be back-projected).
    • Orientation: Upright / Erect (positive magnification $m = +2.00$).
    • Size: Magnified / Enlarged ($|m| = 2.00$; image is twice the height of the object).