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10 Multiple Choice Questions on phase difference to path difference conversion formula:

Topic: Wave Optics, Waves

$$\Delta \phi = \frac{2\pi}{\lambda} \Delta x$$

Question 1

What is the correct mathematical relationship between phase difference ($\Delta \phi$) and path difference ($\Delta x$) for a wave of wavelength $\lambda$?

A) $\Delta \phi = \frac{\lambda}{2\pi} \Delta x$

B) $\Delta \phi = \frac{2\pi}{\lambda} \Delta x$

C) $\Delta \phi = 2\pi \lambda \Delta x$

D) $\Delta \phi = \frac{2\pi \lambda}{\Delta x}$

• Answer: B

Question 2

If the path difference between two interfering light waves is exactly equal to one full wavelength ($\lambda$), what is their phase difference in radians?

A) $\pi$

B) $\frac{\pi}{2}$

C) $2\pi$

D) $4\pi$

• Answer: C

Question 3

A path difference of $\frac{\lambda}{2}$ corresponds to which of the following phase differences?

A) $90^\circ$ ($\frac{\pi}{2}$ rad)

B) $180^\circ$ ($\pi$ rad)

C) $270^\circ$ ($\frac{3\pi}{2}$ rad)

D) $360^\circ$ ($2\pi$ rad)

• Answer: B

Question 4

For two points on a traveling wave, the phase difference is found to be $\frac{\pi}{3}\text{ rad}$. If the wavelength of the wave is $60\text{ cm}$, what is the path difference between these two points?

A) $5\text{ cm}$

B) $10\text{ cm}$

C) $20\text{ cm}$

D) $30\text{ cm}$

• Answer: B
• Solution: $\Delta x = \frac{\lambda}{2\pi} \Delta \phi = \frac{60}{2\pi} \times \frac{\pi}{3} = \frac{60}{6} = 10\text{ cm}$.

Question 5

In the formula $\Delta \phi = \frac{2\pi}{\lambda} \Delta x$, what does the term $\frac{2\pi}{\lambda}$ physically represent?

A) Wave velocity ($v$)

B) Angular frequency ($\omega$)

C) Propagation constant / Wave number ($k$)

D) Time period ($T$)

• Answer: C

Question 6

If the path difference between two monochromatic light waves arriving at a point is zero ($\Delta x = 0$), their phase difference will be:

A) $\pi\text{ rad}$

B) $2\pi\text{ rad}$

C) $0\text{ rad}$

D) Dependent on the wavelength of the light

• Answer: C

Question 7

Two waves having a path difference of $\Delta x = \frac{3\lambda}{4}$ will have a phase difference equal to:

A) $\frac{\pi}{2}\text{ rad}$

B) $\pi\text{ rad}$

C) $\frac{3\pi}{2}\text{ rad}$

D) $3\pi\text{ rad}$

• Answer: C

Question 8

In a medium where the wavelength of a light wave shrinks to half of its vacuum value ($\lambda’ = \frac{\lambda_0}{2}$), how does the phase difference change for a fixed path difference $\Delta x$?

A) It becomes half.

B) It doubles.

C) It quadruples.

D) It remains unchanged.

• Answer: B

Question 9

Which of the following conditions for path difference ($\Delta x$) corresponds to a phase difference of $\Delta \phi = 3\pi\text{ rad}$?

A) $\Delta x = \lambda$

B) $\Delta x = \frac{3\lambda}{2}$

C) $\Delta x = 3\lambda$

D) $\Delta x = \frac{2\lambda}{3}$

• Answer: B

Question 10

Consider the following statements regarding the relation $\Delta \phi = k \Delta x$:

1. The SI unit of phase difference $\Delta \phi$ is radians.
2. For a constant path difference, higher frequency waves (shorter wavelengths) will yield a larger phase difference.

Which of the statements is/are correct?

A) Only 1

B) Only 2

C) Both 1 and 2

D) Neither 1 nor 2

• Answer: C