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(d) 6π\text{(d) } 6\pi

Step-by-step solution

5 steps
  1. Identify the formula for phase difference in Young's double-slit experiment.

    The phase difference (Δϕ\Delta \phi) between two waves is related to their path difference (Δx\Delta x) by the formula:

    Δϕ=2πλΔx\Delta \phi = \frac{2\pi}{\lambda} \Delta x

    where λ\lambda is the wavelength of light.

  2. Determine the path difference for a bright fringe.

    For a bright fringe (constructive interference), the path difference is an integer multiple of the wavelength:

    Δx=mλ\Delta x = m \lambda

    where mm is the order of the bright fringe (m=0,1,2,…m = 0, 1, 2, \dots).

  3. Find the order of the third bright fringe.

    The central bright fringe corresponds to m=0m=0.
    The first bright fringe corresponds to m=1m=1.
    The second bright fringe corresponds to m=2m=2.
    The third bright fringe corresponds to m=3m=3.
    So, for the third bright fringe, m=3m=3.

  4. Calculate the path difference for the third bright fringe.

    Using m=3m=3, the path difference is:

    Δx=3λ\Delta x = 3\lambda
  5. Calculate the phase difference for the third bright fringe.

    Substitute Δx=3λ\Delta x = 3\lambda into the phase difference formula:

    Δϕ=2πλ(3λ)\Delta \phi = \frac{2\pi}{\lambda} (3\lambda) Δϕ=6π\Delta \phi = 6\pi

    The wavelength λ=6000 A˚\lambda = 6000 \text{ Å} is given but is not required for this calculation as it cancels out.

    The final answer is (d) 6π\text{(d) } 6\pi

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