Two parallel slits are illuminated by light composed of two wavelengths. Wavelength A is 564 nm and the other is wavelength B and is unknown. On a viewing screen, the light of wavelength A produces a third-order bright fringe at the same place where the light with wavelength B produces its fourth dark fringe. The fringes are counted relative to the central or zeroth-order bright fringe. What is the unknown wavelength?

The unknown wavelength is 376 nm

Explanation:

We are given two parallel slits that are illuminated by light composed of two wavelengths, λa = 564 nm and the other λb which is unknown.

On a viewing screen, the light whose wavelength is known produces its third dark fringe at the same place where the light with wavelength B produces its fourth dark fringe.

For the wavelength A: (bright fringes)

sinθ = mλa/d

where m = 3

sinθ = 3λa/d

For the wavelength B: (dark fringes)

sinθ = (m + ½)λb/d

where m = 4

sinθ = (4 + ½)λb/d

Since we both fringes are produced at the same place,

3λa/d = (4 + ½)λb/d

d cancels out

3λa = (4 + ½)λb

λb = 3λa/(4 + ½)

λb = 3(564)/(4.5)

λb = 376 nm

Therefore, the other wavelength is 376 nm.

i think it because equal matrices have equal dimensions, only square matrices can be symmetric. for all indices and . every square diagonal matrix is symmetric, since all off-diagonal elements are zero. in linear algebra, a real symmetric matrix represents a self-adjoint operator over a real inner product space.