Phasor Addition and the Mathematics of Interference
Representing each contributing wave as a rotating arrow turns interference into vector addition, so intensity comes from the length of the resultant. Two-slit, multi-slit and single-slit patterns all follow from the same picture.
What a learner can do afterwards
- Adds two or more waves of equal frequency with a phasor diagram and reads off the resultant amplitude
- Derives the intensity pattern of a two-slit interference figure
- Explains how the pattern sharpens as the number of slits increases
1 · Read
Give each wave an arrow called a phasor. Its length is the wave amplitude and its angle is the wave phase. To add waves of equal frequency, lay the arrows head to tail: the arrow from the first tail to the last head is the resultant, and its length is the total amplitude.
Detectors feel energy, which grows as amplitude squared, so brightness goes as the squared length of the resultant. For two slits spaced d apart, the paths to a screen point differ by d times sin theta. Bright fringes sit where that difference is a whole number m of wavelengths, and the fringe spacing on a far screen is wavelength times screen distance over slit spacing.
Try three equal waves whose phases step by 120 degrees. Their three arrows close into a triangle, so the resultant is zero and the screen goes dark there. Put the same three waves in step instead, and the amplitude triples while the brightness grows ninefold.
With many slits the arrows line up fully only at the main maxima, and elsewhere they curl toward zero, so the bright fringes get sharper and narrower. No energy is lost in the dark zones: it is all piled into the bright ones.
Draw each wave as an arrow, add the arrows, and square the resultant length to get the brightness.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.