Question

The condition for bright ring in the Newton's Ring arrangement is (where 't' is thickness of film, m is order and \(\lambda\) is wavelength):

• A. \( 2t = m\lambda + \frac{\lambda}{2} \)
• B. \( t = m\lambda - \frac{\lambda}{2} \)
• C. \( 2t = \frac{(2m-1)\lambda}{2} \)
• D. \( t = \frac{m\lambda}{2} \)

Hint:

For interference in thin films, always check for phase changes on reflection. A reflection from a denser medium adds \(\lambda/2\) to the path difference. This is why in Newton's rings (and soap bubbles), the conditions for bright and dark fringes are "swapped" compared to what you might expect from the geometrical path difference alone. The center of Newton's rings (where t=0) is dark because the \(\lambda/2\) phase shift causes destructive interference.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Newton's rings are formed due to interference between light waves reflected from the top and bottom surfaces of a thin air film trapped between a plano-convex lens and a flat glass plate. A key aspect is the phase change of \(\pi\) (or path difference of \(\lambda/2\)) that occurs upon reflection from a denser medium.
Step 2: Key Formula or Approach:
1. Light traveling from a rarer medium (air) reflects off the surface of a denser medium (the bottom glass plate). This reflection introduces a phase shift of \(\pi\), which is equivalent to an extra path difference of \(\lambda/2\). 2. The reflection from the top surface (the bottom of the lens) is from a rarer medium (glass) to a denser medium (air), so there is no phase shift here. 3. The geometrical path difference for a wave that travels through the film of thickness 't' and back is \(2t\) (assuming near-normal incidence). 4. The total optical path difference is \( \Delta = 2t + \frac{\lambda}{2} \). 5. For constructive interference (a bright fringe), the total path difference must be an integer multiple of the wavelength: \( \Delta = m\lambda \), where \(m = 1, 2, 3, ...\).
Step 3: Detailed Explanation:
Setting the condition for a bright fringe:
\[ \text{Total path difference} = m\lambda \] \[ 2t + \frac{\lambda}{2} = m\lambda \] Solving for \(2t\):
\[ 2t = m\lambda - \frac{\lambda}{2} \] \[ 2t = \left(m - \frac{1}{2}\right)\lambda \] \[ 2t = \frac{(2m-1)\lambda}{2} \] This matches option (C). Option (A) is also mathematically equivalent, but option (C) is a more standard representation.
Step 4: Final Answer:
The condition for a bright ring is \( 2t = \frac{(2m-1)\lambda}{2} \).