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Electric Charges and Fields
- Electric Charge
- Conductors and Insulators
- Basic Properties of Electric Charge
- Coulomb’s Law
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- Electric Field
- Electric Field Due to a System of Charges
- Physical Significance of Electric Field
- Electric Field Lines
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- Dipole in a Uniform External Field
- Continuous Charge Distribution
- Gauss’s Law
- Application of Gauss' Law
Electrostatics
Electrostatic Potential and Capacitance
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- The Parallel Plate Capacitor
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Ray Optics and Optical Instruments
- Ray Optics Or Geometrical Optics
- Reflection of Light by Spherical Mirrors
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- Focal Length of Spherical Mirrors
- Mirror Equation of Spherical Mirrors
- Refraction of Light
- Total Internal Reflection
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- Concept of Wave Optics
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Communication Systems
Dual Nature of Radiation and Matter
- Understanding Dual Nature of Radiation and Matter
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The Special Theory of Relativity
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Semiconductor Electronics - Materials, Devices and Simple Circuits
- Concept of Semiconductor Electronics
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- Intrinsic Semiconductor
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- Overview: Semiconductor Electronics
Communication Systems
- Detection of Amplitude Modulated Wave
- Production of Amplitude Modulated Wave
- Basic Terminology Used in Electronic Communication Systems
- Sinusoidal Waves
- Modulation and Its Necessity
- Amplitude Modulation (AM)
- Need for Modulation and Demodulation
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- Bandwidth of Transmission Medium
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The Special Theory of Relativity
- The Special Theory of Relativity
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- Dynamics at Large Velocity
- Energy and Momentum
- The Ultimate Speed
- Twin Paradox
Introduction
When a parallel beam of paraxial light strikes a spherical mirror, the reflected rays do not scatter randomly — they either converge to a single point (concave mirror) or appear to diverge from a single point (convex mirror). This special point, and the distance to it from the mirror surface, defines what we call the focal length.
Focal length is arguably the most important single parameter describing a spherical mirror. The mirror formula, magnification, and image position — all depend on it. The key result proved in this module is:
That is, the focal length of any spherical mirror equals exactly half its radius of curvature.
Definition: Principal Focus
The point F on the principal axis where a parallel paraxial beam of light converges (or appears to diverge from) after reflection is called the Principal Focus of the mirror.
Definition: Focal Plane
The plane perpendicular to the principal axis passing through the principal focus F is called the Focal Plane of the mirror.
Definition: Focal Length
The distance between the Principal Focus F and the Pole P of the mirror is called the Focal Length, denoted by f.
f = \[\overline {PF}\]
Derivation of f = R/2
Setup
Consider a concave spherical mirror with:
- P = Pole of the mirror
- C = Centre of curvature (radius of curvature = R = CP)
- F = Principal focus
- A ray parallel to the principal axis strikes the mirror at point M
Step 1: Identify the normal at M
The normal at any point on a spherical mirror passes through the centre of curvature C. Therefore, CM is the normal at M.
Step 2: Apply the law of reflection
Let θ = angle of incidence (between incident ray and CM).
By the law of reflection, the angle of reflection = θ.
Therefore:
- ∠MCP = θ (angle at C in triangle MCF)
- ∠MFP = 2θ (exterior angle of triangle MCF)ncert
Step 3: Set up trigonometric relations
Drop a perpendicular MD from M to the principal axis. Then:
- tan θ = \[\frac {MD}{CD}\] and tan 2θ = \[\frac {MD}{FD}\]
Step 4: Apply small-angle (paraxial) approximation
For paraxial rays, θ is very small. Hence:
- tan θ ≈ θ and tan 2θ ≈ 2θ
Substituting:
- \[\frac {MD}{FD}\] = 2 ⋅ \[\frac {MD}{CD}\]
- FD = \[\frac {CD}{2}\]
Step 5: Apply paraxial approximation to distances
For paraxial rays, M is very close to P, so D is very close to P:
- FD ≈ FP = f and CD ≈ CP = R
Step 6: Final result
Substituting into equation (1):
- f = \[\frac {R}{2}\]
This result holds for both concave and convex mirrors.
