Delta phi = (2 pi / lambda) Delta x
Phase difference equals two pi over wavelength, multiplied by path difference.
Phase difference corresponding to a given path difference.
- Delta phi
- phase difference (rad)
- lambda
- wavelength (m)
- Delta x
- path difference (m)
Use when — Converting between path difference and phase difference for the same wave.
Common trap — Mixing path difference measured in wavelengths with phase difference measured in radians without converting.
d sin(theta) = n lambda
Slit separation times sine of the angle equals an integer multiple of the wavelength.
Condition for a bright fringe in Young's double slit experiment.
- d
- slit separation (m)
- theta
- angle from the central axis (rad)
- n
- fringe order, integer (dimensionless)
- lambda
- wavelength (m)
Use when — Two coherent slits with a well-defined slit separation illuminated by one wavelength.
Common trap — Using this for a single-slit diffraction pattern, which follows a different condition.
beta = lambda D / d
Fringe width equals wavelength times screen distance, divided by slit separation.
Fringe width on a distant screen in Young's double slit experiment.
- beta
- fringe width (m)
- D
- slit-to-screen distance (m)
- d
- slit separation (m)
Use when — Screen distance is much greater than slit separation, so the small-angle approximation applies.
Common trap — Using this formula when the screen is close to the slits, where the small-angle approximation breaks down.
I = I1 + I2 + 2 sqrt(I1 I2) cos(delta)
Resultant intensity equals the sum of the two intensities plus twice the square root of their product times cosine of the phase difference.
Resultant intensity from two coherent waves with a constant phase difference.
- I1, I2
- individual intensities (W/m^2)
- delta
- constant phase difference between the two waves (rad)
Use when — The two sources are coherent, so delta does not vary randomly with time.
Common trap — Applying this to incoherent sources, where intensities simply add without the interference term.
I_max/I_min = (sqrt(I1) + sqrt(I2))^2 / (sqrt(I1) - sqrt(I2))^2
The ratio of maximum to minimum intensity equals the square of the sum of the square roots of the two intensities, divided by the square of their difference.
Ratio of maximum to minimum intensity in an interference pattern from two coherent sources.
- I_max
- intensity at a bright fringe (W/m^2)
- I_min
- intensity at a dark fringe (W/m^2)
Use when — The two coherent sources have possibly unequal individual intensities.
Common trap — Assuming dark fringes are completely dark when the two source intensities are unequal.
a sin(theta) = n lambda
Slit width times sine of the angle equals an integer multiple of the wavelength, for diffraction minima.
Condition for a dark fringe (minimum) in single-slit diffraction.
- a
- slit width (m)
- theta
- angle from the central axis (rad)
- n
- minimum order, nonzero integer (dimensionless)
Use when — Single narrow slit diffraction, locating the dark fringes.
Common trap — Using this condition and expecting it to mark bright fringes; here it marks the minima, unlike the double-slit condition.
width = 2 lambda D / a
The width of the central maximum equals two times wavelength times screen distance, divided by slit width.
Angular width of the central maximum in single-slit diffraction, expressed as a linear width on a distant screen.
- D
- slit-to-screen distance (m)
- a
- slit width (m)
Use when — Small-angle approximation holds and the screen is far from the slit.
Common trap — Assuming a narrower slit always gives a narrower central maximum; the opposite happens in diffraction.
theta_min = 1.22 lambda / D
Minimum resolvable angle equals 1.22 times wavelength, divided by aperture diameter.
Minimum angular separation two point objects can have and still be resolved, set by aperture diffraction.
- theta_min
- minimum resolvable angular separation (rad)
- D
- aperture diameter (m)
Use when — Resolving power of a telescope or the objective of a microscope, limited by diffraction at the aperture.
Common trap — Treating a smaller angular limit as meaning the instrument resolves more, when a smaller angle actually means finer resolving ability.
I = I0 cos^2(theta)
Transmitted intensity equals incident intensity times cosine squared of the angle between the two transmission axes.
Transmitted intensity through an analyser Polaroid, given plane-polarised incident light.
- I0
- intensity of the incident plane-polarised light (W/m^2)
- theta
- angle between the polariser and analyser transmission axes (rad)
Use when — The light entering the second Polaroid is already plane-polarised, from a first Polaroid or another polarising process.
Common trap — Applying Malus law directly to unpolarised light entering the first Polaroid instead of the light leaving it.
tan(theta_B) = n21
Tangent of the Brewster angle equals the relative refractive index of the second medium with respect to the first.
Brewster angle for complete polarisation of the reflected ray at a boundary.
- theta_B
- Brewster angle of incidence (rad)
- n21
- refractive index of the second medium relative to the first (dimensionless)
Use when — Finding the angle of incidence at which reflected light is completely plane polarised.
Common trap — Forgetting that the reflected and refracted rays are perpendicular only at exactly this angle.
