A group of students conducted a series of trials to study the interference patterns of lig… — ACT

A group of students conducted a series of trials to study the interference patterns of light using a double-slit setup. They measured the fringe spacing, $w$ (the distance between adjacent bright bands on a screen). The relationship between the fringe spacing and the experimental parameters is given by:

w = \frac{\lambda L}{d}

where $\lambda$ is the wavelength of the light source, $L$ is the distance from the slits to the screen, and $d$ is the distance between the two slits.

Match each change in the experimental setup to its corresponding effect on the fringe spacing ( $w$ ).

Doubling the slit separation ( $d$ ) while keeping all other variables constant.The fringe spacing ( $w$ ) is halved because of an inverse proportionality.
Doubling the wavelength of the light ( $\lambda$ ) while keeping all other variables constant.The fringe spacing ( $w$ ) is doubled because of a direct proportionality.
Doubling both the slit separation ( $d$ ) and the distance to the screen ( $L$ ) simultaneously.The fringe spacing ( $w$ ) remains unchanged because the direct and inverse proportionalities cancel each other out.

Answer

To solve this, match the physical changes to their mathematical consequences: doubling the slit separation (

d

) halves the fringe spacing (

w

); doubling the wavelength (

\lambda

) doubles the fringe spacing (

w

); and doubling both the slit separation (

d

) and the distance to the screen (

L

) keeps the fringe spacing (

w

) unchanged.

The correct matches are based on the algebraic relationship

w = \frac{\lambda L}{d}

. Doubling a variable in the numerator (

\lambda

) doubles the value of

w

due to direct proportionality. Doubling a variable in the denominator (

d

) halves the value of

w

due to inverse proportionality. Doubling both simultaneously cancels the changes out (

\frac{2}{2} = 1

), keeping

w

constant.

Step-by-Step Solution

Identify the proportional relationships for each variable in the equation

w = \frac{\lambda L}{d}

w

is directly proportional to

\lambda

and

L

, and inversely proportional to

d

This allows us to determine how changing each variable independently affects the value of

w

Analyze the effect of doubling the slit separation (

d

w' = \frac{\lambda L}{2d} = \frac{1}{2}w

Because

w

is inversely proportional to

d

, doubling

d

must result in halving

w

Analyze the effect of doubling the wavelength (

\lambda

w' = \frac{2\lambda L}{d} = 2w

Because

w

is directly proportional to

\lambda

, doubling

\lambda

must result in doubling

w

Analyze the effect of simultaneously doubling the slit separation (

d

) and the screen distance (

L

w' = \frac{\lambda (2L)}{2d} = \frac{2}{2}w = w

The direct proportionality factor of 2 from

L

and the inverse proportionality factor of 2 from

d

cancel each other out, leaving the fringe spacing unchanged.

Key Concept

Direct and inverse proportionality in algebraic equations

Estimated Time:1m 30s

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