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: Concept Introduction:
Newton's rings arise from the interference of light waves reflecting from the two surfaces of a thin air film. This film is situated between a plano-convex lens and a flat glass plate. A critical factor is the \(\pi\) phase shift (or \(\lambda/2\) path difference) encountered during reflection from a denser medium.
Step 2: Core Principles:
1. Reflection of light from a rarer medium (air) to a denser medium (glass plate) induces a \(\pi\) phase shift, equivalent to a \(\lambda/2\) path difference.2. Reflection from the lens's bottom surface (glass to air) does not cause a phase shift.3. The geometric path difference for light traversing the film of thickness 't' and returning is approximately \(2t\) (for near-normal incidence).4. The total optical path difference is \( \Delta = 2t + \frac{\lambda}{2} \).5. For constructive interference (bright fringe), the total path difference must equal an integer multiple of the wavelength: \( \Delta = m\lambda \), with \(m = 1, 2, 3, ...\).
Step 3: Derivation:
Applying the condition for constructive interference:
\[ \text{Total path difference} = m\lambda \]\[ 2t + \frac{\lambda}{2} = m\lambda \]Rearranging to solve for \(2t\):
\[ 2t = m\lambda - \frac{\lambda}{2} \]\[ 2t = \left(m - \frac{1}{2}\right)\lambda \]\[ 2t = \frac{(2m-1)\lambda}{2} \]This result aligns with option (C). Option (A) is mathematically equivalent, but (C) is the standard representation.
Step 4: Conclusion:
The condition for a bright ring in Newton's rings is given by \( 2t = \frac{(2m-1)\lambda}{2} \).