Light-path length difference

[Foxhound101]

Homework Statement

Two narrow slits are 0.12 mm apart. Light of wavelength 550 nm illuminates the slits, causing an interference pattern on a screen 1.0 m away. Light from each slit travels to the m=1 maximum on the right side of the central maximum.

Part A -
How much farther did the light from the left slit travel than the light from the right slit?
Express your answer using two significant figures.

Homework Equations

r=dsin(theta)
(theta)_m = m*(lambda/d)
y=L*tan(theta)

y_m = (m*lambda*L)/d

The Attempt at a Solution

I don't understand how to do these problems...

theta_m = (m*lambda*)/d
theta_m = (1*(5.5*10^-7m)/(1m)
theta_m = 5.5*10^-7


path length difference = dsin(theta)
so...
r = d*sin(theta)
r = 1m *sin(5.5*10^-7)
r = 9.599^-9m
r = 9.6nm
That doesn't appear to be the correct answer(Unless MasteringPhysics is wrong). Sadly, I don't know if I did the right steps or used the correct equations.

Any help is appreciated.

 

[alphysicist]

Hi Foxhound101,

Homework Statement

Two narrow slits are 0.12 mm apart. Light of wavelength 550 nm illuminates the slits, causing an interference pattern on a screen 1.0 m away. Light from each slit travels to the m=1 maximum on the right side of the central maximum.

Part A -
How much farther did the light from the left slit travel than the light from the right slit?
Express your answer using two significant figures.

Homework Equations

r=dsin(theta)
(theta)_m = m*(lambda/d)
y=L*tan(theta)

y_m = (m*lambda*L)/d

The Attempt at a Solution

I don't understand how to do these problems...

theta_m = (m*lambda*)/d
theta_m = (1*(5.5*10^-7m)/(1m)
theta_m = 5.5*10^-7

Remember that this is really:

$$\sin\theta=\frac{m\lambda}{d}$$

The approximation you are using ($\theta=\frac{m\lambda}{d}$) is fine since the angle is small enough, but remember that this approximation is true if the angle is measured in radians. So the angle you found is $5.5\times 10^{-7}\mbox{ rad}$.

path length difference = dsin(theta)
so...
r = d*sin(theta)
r = 1m *sin(5.5*10^-7)
r = 9.599^-9m

This number was calculated with the angle measure set to degrees, not radians.

 

[baba89]

Hello, thank you for reaching out for help with this problem. I can see that you have attempted to use the equations for finding the angle of diffraction and the path length difference, but it seems that you may have mixed up some of the values. Let me walk you through the correct steps for finding the path length difference in this situation.

First, we need to find the angle of diffraction for the m=1 maximum on the right side of the central maximum. This can be done using the equation theta = m(lambda)/d, where m is the order of the maximum (in this case, m=1), lambda is the wavelength of light (550 nm), and d is the distance between the two slits (0.12 mm = 0.00012 m). Plugging in these values, we get:

theta = (1)*(550*10^-9 m)/(0.00012 m)
theta = 4.58 radians

Next, we can use the equation for the path length difference, r = d*sin(theta), to find the difference in path length between the light from the left slit and the right slit. Plugging in the value for theta that we just calculated, we get:

r = (0.00012 m)*sin(4.58 radians)
r = 5.5*10^-7 m

This is the same value you obtained, but your units were incorrect. Remember that the value for theta should be in radians, not meters. So the correct answer for the path length difference is 5.5*10^-7 m, or 0.55 micrometers.

I hope this explanation helps you understand the steps involved in solving this problem. Keep practicing and don't hesitate to ask for clarification if you come across any difficulties. Good luck with your studies!