The 4th nodal line is observed 3.6 cm from the centreline in a ripple tank containing 2 point sources. The sources are 1.60 cm apart and the screen is 120.0 cm away. What is the wavelength?

Question

BlackTom AI · Accepted Answer

We need to use the standard fringe formula for two-slit (or two-source) interference observed on a screen at distance L: y_m = (m + 1/2) * lambda * L / d, where y_m is the distance from the central (zero-order) bright or dark line to the m-th nodal (dark) line, d is the separation between the two sources, and lambda is the wavelength.

First, identify the order of the nodal line given. The problem states the 4th nodal line is observed at a distance of 3.6 cm from the centreline. The nth nodal (dark) line corresponds to m = n − 1 in the formula, so for the 4th nodal line we take m = 3.

Compute lambda from the given data using the formula:
- y = (m + 1/2) * lambda * L / d
- 3.6 cm = (3 + 1/2) * lambda * 120.0 cm / 1.60 cm
- 3.6 = 3.5 * lambda * 120 / 1.60 (in consistent units; distance units cancel)
- lambda = (3.6 * 1.60) / (3.5 * 120)
- lambda = 5.76 / 420 ≈ 0.013714... cm = 0.00013714 m = 0.137 mm

Now compare this calculated value to each option:

- Option a: 0.14 mm. This is very close to 0.137 mm. The slight difference arises from rounding during intermediate steps; it is a consistent and reasonable match to the computed wavelength using the given measurements.

- Option b: 1.4 mm. This is about ten times larger than the computed value. Such a discrepancy would imply a far larger wavelength than the geometry suggests, indicating a misapplication of the fringe formula or a misreading of the order when writing y_m.

- Option c: 0.23 mm. This is larger than the calculated 0.137 mm. It could result from using a different fringe order (e.g., treating the line as (m + 1) rather than (m + 1/2)) or miscounting the nodal line, but would not align with the standard y_m = (m + 1/2) lambda L / d relation for dark fringes.

- Option d: 0.096 mm. This value is smaller than the computed wavelength. If one were to plug this back into the formula, the predicted y would be smaller than 3.6 cm for the same L and d, so this option would not reproduce the given geometry under the correct model.

In summary, the correct interpretation uses the half-integer offset for the dark fringe positions and the given order m = 3, which yields a wavelength near 0.137 mm, matching option a within typical rounding. The other options arise from either misidentifying the fringe order, neglecting the 1/2 lambda offset, or inconsistent unit handling, and hence do not fit the given measurements.

SPH4U - Physics 12 (2025-26) - A

The 4th nodal line is observed 3.6 cm from the centreline in a ripple tank containing 2 point sources. The sources are 1.60 cm apart and the screen is 120.0 cm away. What is the wavelength?

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