Formula Sheet: Light and Electromagnetic Radiation

Electromagnetic Waves and Light Properties

Wave Properties of Light

Speed of Light in Vacuum \[ c = 3.00 \times 10^8 \text{ m/s} \]

• c = speed of light in vacuum (constant)
• This is the maximum speed at which all electromagnetic radiation travels in vacuum

Wave Equation \[ c = f \lambda \]

• c = speed of light (3.00 × 10⁸ m/s)
• f = frequency (Hz or s^-1)
• λ = wavelength (m)
• This fundamental relationship applies to all electromagnetic radiation
• Frequency and wavelength are inversely proportional

Speed of Light in a Medium \[ v = \frac{c}{n} \]

• v = speed of light in the medium (m/s)
• c = speed of light in vacuum (3.00 × 10⁸ m/s)
• n = index of refraction of the medium (dimensionless)
• Light slows down when traveling through any medium (n > 1)
• For vacuum, n = 1; for air, n ≈ 1.00; for water, n ≈ 1.33; for glass, n ≈ 1.5

Wavelength in a Medium \[ \lambda_{\text{medium}} = \frac{\lambda_{\text{vacuum}}}{n} \]

• Wavelength decreases when light enters a denser medium
• Frequency remains constant when light crosses medium boundaries

Energy and Photons

Photon Energy \[ E = hf \]

• E = energy of a photon (J)
• h = Planck's constant = 6.626 × 10^-34 J·s
• f = frequency (Hz)
• Energy is directly proportional to frequency

Photon Energy in Terms of Wavelength \[ E = \frac{hc}{\lambda} \]

• E = energy of a photon (J)
• h = Planck's constant (6.626 × 10^-34 J·s)
• c = speed of light (3.00 × 10⁸ m/s)
• λ = wavelength (m)
• Energy is inversely proportional to wavelength
• Shorter wavelengths (like UV) have higher energy; longer wavelengths (like IR) have lower energy

Energy Using Electron Volts \[ E \text{ (eV)} = \frac{1240 \text{ eV·nm}}{\lambda \text{ (nm)}} \]

• Useful approximation for MCAT calculations
• 1 eV = 1.6 × 10^-19 J
• Wavelength must be in nanometers (nm)

Photon Momentum \[ p = \frac{E}{c} = \frac{h}{\lambda} \]

• p = momentum of a photon (kg·m/s)
• E = energy (J)
• h = Planck's constant (6.626 × 10^-34 J·s)
• λ = wavelength (m)
• Photons have momentum despite having zero rest mass

Electromagnetic Spectrum

Order of Electromagnetic Spectrum (Increasing Frequency/Energy)

1. Radio waves (lowest energy, longest wavelength)
2. Microwaves
3. Infrared (IR)
4. Visible light (ROYGBIV: Red, Orange, Yellow, Green, Blue, Indigo, Violet)
5. Ultraviolet (UV)
6. X-rays
7. Gamma rays (highest energy, shortest wavelength)

Visible Light Spectrum

• Red: ~700 nm (lowest energy, longest wavelength)
• Orange: ~600 nm
• Yellow: ~580 nm
• Green: ~550 nm
• Blue: ~470 nm
• Violet: ~400 nm (highest energy, shortest wavelength)
• Visible spectrum: approximately 400-700 nm

Reflection

Law of Reflection

\[ \theta_i = \theta_r \]

• θ_i = angle of incidence (measured from the normal)
• θ_r = angle of reflection (measured from the normal)
• Both angles are measured from the normal (perpendicular) to the surface
• The incident ray, reflected ray, and normal all lie in the same plane

Mirror Equation

\[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]

• f = focal length (m or cm)
• d_o = object distance from mirror (m or cm)
• d_i = image distance from mirror (m or cm)
• Applies to both concave and convex mirrors

Sign Conventions for Mirrors:

• f is positive for concave (converging) mirrors
• f is negative for convex (diverging) mirrors
• d_i is positive for real images (in front of mirror)
• d_i is negative for virtual images (behind mirror)
• d_o is always positive for real objects

Magnification

\[ m = -\frac{d_i}{d_o} = \frac{h_i}{h_o} \]

• m = magnification (dimensionless)
• d_i = image distance (m or cm)
• d_o = object distance (m or cm)
• h_i = image height (m or cm)
• h_o = object height (m or cm)

Sign Conventions for Magnification:

• m > 0: upright image
• m <>: inverted image
• |m| > 1: enlarged/magnified image
• |m| <>: reduced/diminished image

Focal Length and Radius of Curvature

\[ f = \frac{R}{2} \]

• f = focal length (m or cm)
• R = radius of curvature (m or cm)
• The focal point is located halfway between the center of curvature and the mirror surface
• For concave mirrors: f and R are positive
• For convex mirrors: f and R are negative

Power of a Mirror

\[ P = \frac{1}{f} \]

• P = power (diopters, D = m^-1)
• f = focal length (must be in meters)
• Positive power indicates a converging mirror
• Negative power indicates a diverging mirror

Refraction

Snell's Law

\[ n_1 \sin\theta_1 = n_2 \sin\theta_2 \]

• n₁ = index of refraction of first medium (dimensionless)
• θ₁ = angle of incidence in first medium (measured from normal)
• n₂ = index of refraction of second medium (dimensionless)
• θ₂ = angle of refraction in second medium (measured from normal)
• Light bends toward the normal when entering a denser medium (n₂ > n₁)
• Light bends away from the normal when entering a less dense medium (n₂ <>₁)

Critical Angle and Total Internal Reflection

\[ \sin\theta_c = \frac{n_2}{n_1} \]

• θ_c = critical angle
• n₁ = index of refraction of the denser medium (where light originates)
• n₂ = index of refraction of the less dense medium
• This formula only applies when n₁ > n₂ (light going from denser to less dense medium)
• When θ₁ ≥ θ_c, total internal reflection occurs (no refraction, 100% reflection)
• Applications: fiber optics, prisms, diamonds

Lenses - Thin Lens Equation

\[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]

• f = focal length (m or cm)
• d_o = object distance from lens (m or cm)
• d_i = image distance from lens (m or cm)
• Same form as mirror equation but different sign conventions

Sign Conventions for Lenses:

• f is positive for converging lenses (convex)
• f is negative for diverging lenses (concave)
• d_i is positive for real images (opposite side of lens from object)
• d_i is negative for virtual images (same side of lens as object)
• d_o is positive for real objects (always for simple problems)

Magnification for Lenses

\[ m = -\frac{d_i}{d_o} = \frac{h_i}{h_o} \]

• Same formula as mirrors
• m > 0: upright image
• m <>: inverted image

Lensmaker's Equation

\[ \frac{1}{f} = (n-1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right) \]

• f = focal length of the lens
• n = index of refraction of the lens material
• R₁ = radius of curvature of first surface
• R₂ = radius of curvature of second surface
• R is positive if the center of curvature is on the side where light exits
• R is negative if the center of curvature is on the side where light enters

Power of a Lens

\[ P = \frac{1}{f} \]

• P = power (diopters, D = m^-1)
• f = focal length (must be in meters)
• Positive power indicates a converging (convex) lens
• Negative power indicates a diverging (concave) lens

Combined Power of Thin Lenses in Contact \[ P_{\text{total}} = P_1 + P_2 + P_3 + ... \]

• Powers add algebraically when lenses are in contact
• Equivalent to: \( \frac{1}{f_{\text{total}}} = \frac{1}{f_1} + \frac{1}{f_2} + \frac{1}{f_3} + ... \)

Dispersion

Index of Refraction Wavelength Dependence

• Index of refraction varies with wavelength: n = n(λ)
• Generally, shorter wavelengths (violet) have higher n than longer wavelengths (red)
• This causes white light to separate into colors when passing through a prism
• Dispersion: separation of light into component colors due to wavelength-dependent refraction

Interference and Diffraction

Double-Slit Interference (Young's Experiment)

Condition for Bright Fringes (Constructive Interference) \[ d\sin\theta = m\lambda \]

• d = distance between slits (m)
• θ = angle from central axis to the fringe
• m = order of the bright fringe (m = 0, ±1, ±2, ±3, ...)
• λ = wavelength of light (m)
• m = 0 corresponds to the central bright fringe
• Path difference = mλ for constructive interference

Condition for Dark Fringes (Destructive Interference) \[ d\sin\theta = \left(m + \frac{1}{2}\right)\lambda \]

• m = 0, ±1, ±2, ±3, ...
• Path difference = (m + ½)λ for destructive interference
• Dark fringes occur between bright fringes

Fringe Spacing (Small Angle Approximation) \[ y = \frac{m\lambda L}{d} \]

• y = distance from central maximum to mth bright fringe
• L = distance from slits to screen
• d = slit separation
• λ = wavelength
• m = fringe order
• Valid when θ is small (typically when L >> d)

Distance Between Adjacent Bright Fringes \[ \Delta y = \frac{\lambda L}{d} \]

• Spacing between consecutive bright fringes is uniform
• Larger wavelength → larger spacing
• Larger slit separation → smaller spacing

Single-Slit Diffraction

Condition for Dark Fringes (Minima) \[ a\sin\theta = m\lambda \]

• a = width of the single slit (m)
• θ = angle to the dark fringe
• m = order of the minimum (m = ±1, ±2, ±3, ...; note m ≠ 0)
• λ = wavelength (m)
• Central maximum is at m = 0, which is bright
• First minimum occurs at m = ±1

Width of Central Maximum \[ w = \frac{2\lambda L}{a} \]

• w = width of central bright fringe
• L = distance from slit to screen
• a = slit width
• λ = wavelength
• Narrower slits produce wider diffraction patterns

Thin Film Interference

Path Difference in Thin Films \[ 2nt = m\lambda \quad \text{(constructive if no phase change or both surfaces have phase change)} \] \[ 2nt = \left(m + \frac{1}{2}\right)\lambda \quad \text{(constructive if one surface has phase change)} \]

• n = index of refraction of the film
• t = thickness of the film
• λ = wavelength in vacuum
• m = 0, 1, 2, 3, ...

Phase Change Upon Reflection

• Light reflects with a 180° (π) phase change when reflecting off a medium with higher index of refraction (n_reflected > n_incident)
• No phase change when reflecting off a medium with lower index of refraction
• Must account for phase changes at both surfaces of the film

Minimum Thickness for Constructive Interference (One Phase Change) \[ t_{\text{min}} = \frac{\lambda}{4n} \]

• When one surface causes a phase change
• This is for m = 0 (first order)

Polarization

Malus's Law

\[ I = I_0 \cos^2\theta \]

• I = intensity of light after passing through polarizer
• I₀ = intensity of polarized light incident on polarizer
• θ = angle between incident light's polarization direction and polarizer axis
• When θ = 0°, maximum transmission (I = I₀)
• When θ = 90°, no transmission (I = 0)

Unpolarized Light Through First Polarizer \[ I = \frac{I_0}{2} \]

• Unpolarized light passing through a polarizer has its intensity reduced by half
• The transmitted light is now polarized along the polarizer axis

Brewster's Angle \[ \tan\theta_B = \frac{n_2}{n_1} \]

• θ_B = Brewster's angle (angle of incidence)
• n₁ = index of refraction of first medium
• n₂ = index of refraction of second medium
• At Brewster's angle, reflected light is completely polarized perpendicular to the plane of incidence
• The reflected and refracted rays are perpendicular to each other

Photoelectric Effect

Work Function and Threshold Frequency

\[ E_{\text{photon}} = \phi + KE_{\text{max}} \] \[ hf = \phi + KE_{\text{max}} \]

• h = Planck's constant (6.626 × 10^-34 J·s)
• f = frequency of incident light (Hz)
• φ = work function (minimum energy needed to eject electron) (J or eV)
• KE_max = maximum kinetic energy of ejected electron (J or eV)

Threshold Frequency \[ f_0 = \frac{\phi}{h} \]

• f₀ = threshold frequency (minimum frequency to eject electrons)
• If f <>₀, no electrons are ejected regardless of intensity
• If f ≥ f₀, electrons are ejected

Threshold Wavelength \[ \lambda_0 = \frac{c}{f_0} = \frac{hc}{\phi} \]

• λ₀ = maximum wavelength that can eject electrons
• If λ > λ₀, no electrons are ejected

Maximum Kinetic Energy \[ KE_{\text{max}} = \frac{1}{2}mv_{\text{max}}^2 = hf - \phi \]

• m = mass of electron (9.11 × 10^-31 kg)
• v_max = maximum velocity of ejected electron
• Excess photon energy beyond work function becomes kinetic energy

Stopping Potential \[ eV_s = KE_{\text{max}} \] \[ V_s = \frac{hf - \phi}{e} \]

• V_s = stopping potential (voltage needed to stop most energetic electrons)
• e = elementary charge (1.6 × 10^-19 C)
• Stopping potential is independent of light intensity

Key Photoelectric Effect Principles

• Photon energy (not intensity) determines if electrons are ejected
• Intensity determines the number of electrons ejected per second (rate)
• Higher frequency light produces electrons with greater kinetic energy
• Electron emission is instantaneous (no time delay)
• Each photon can eject at most one electron (one-to-one interaction)

Radiation Intensity

Intensity of Electromagnetic Radiation

\[ I = \frac{P}{A} \]

• I = intensity (W/m²)
• P = power (W)
• A = area (m²)
• Intensity is power per unit area

Intensity from a Point Source \[ I = \frac{P}{4\pi r^2} \]

• r = distance from point source
• Intensity follows inverse square law
• Doubling the distance reduces intensity to 1/4

Inverse Square Law \[ \frac{I_1}{I_2} = \frac{r_2^2}{r_1^2} \]

• Intensity is inversely proportional to the square of the distance

Optical Instruments

The Human Eye

Power of the Eye \[ P = P_{\text{cornea}} + P_{\text{lens}} \]

• Total power is the sum of cornea and lens powers
• Most refraction occurs at the cornea
• The lens provides fine-tuning (accommodation)

Near Point

• Normal near point: approximately 25 cm
• Closest distance at which the eye can focus
• Increases with age (presbyopia)

Far Point

• Normal far point: infinity
• Farthest distance at which the eye can focus
• Finite far point indicates myopia (nearsightedness)

Vision Correction

Myopia (Nearsightedness)

• Far point is closer than infinity
• Corrected with diverging (concave) lens
• Lens power is negative

Hyperopia (Farsightedness)

• Near point is farther than 25 cm
• Corrected with converging (convex) lens
• Lens power is positive

Simple Magnifier

\[ m = \frac{25 \text{ cm}}{f} \]

• m = angular magnification
• f = focal length of the lens (cm)
• 25 cm represents the near point of normal vision
• Object should be placed at or just inside the focal point

Compound Microscope

\[ m = m_o \times m_e = -\frac{L \times 25\text{ cm}}{f_o \times f_e} \]

• m = total magnification
• m_o = magnification of objective lens
• m_e = magnification of eyepiece
• L = tube length (distance between focal points of objective and eyepiece)
• f_o = focal length of objective lens
• f_e = focal length of eyepiece
• Negative sign indicates inverted image

Refracting Telescope

\[ m = -\frac{f_o}{f_e} \]

• m = angular magnification
• f_o = focal length of objective lens (long)
• f_e = focal length of eyepiece (short)
• Negative sign indicates inverted image
• Larger f_o and smaller f_e produce greater magnification

Telescope Length \[ L = f_o + f_e \]

• L = length of telescope
• Sum of focal lengths when adjusted for viewing distant objects