A beam of yellow laser light (590 nm) passes through a circular aperture of diameter 5.0 mm. What is the angular width of the central diffraction maximum formed on a screen?

Question

A beam of yellow laser light (590 nm) passes through a circular aperture of diameter 5.0 mm. What is the angular width of the central diffraction maximum formed on a screen?

I can't seem to get this answer right. I don't think I am finding the first part of the problem correctly, so could someone please correct me?

asintheta = .087lamda
so .087 as the first minimum

sintheta = (.087)(590e-9m) / (5.0e-3m) = 1.02e-5
sin-1 (1.02e-5) = 5.88e-4 degrees
(5.88e-4) - (.087) = 8.64e-2 degrees for angular width.

Where did i go wrong and how do I fix i

Answer 1

http://en.wikipedia.org/wiki/Angular_resolution

a sinTheta= 1.22 lambda

1.22 comes from a Bessel function, indicating the distance to the first min.

Answer 2

I just tried the problem with 1.22 and i got -1.21 as my final angular width. However, this is also incorrect.

any ideas where I am going wrong?

Answer 3

I have also tried using the positive of the angular width. But that is wrong, too. I used the same equation as i did above but with the 1.22

Answer 4

To find the angular width of the central diffraction maximum formed on a screen, you need to use the formula for the angular position of the first minimum in the diffraction pattern.

The formula you used, asin(theta) = 0.087 * lambda, is correct for finding the angle at which the first minimum occurs. However, you made a mistake in your calculation of the sine inverse.

Let's correct the calculation:

lambda = 590 nm = 590e-9 m
d = diameter of the aperture = 5.0 mm = 5.0e-3 m

Using the formula asin(theta) = 0.087 * lambda, we can rearrange it to solve for theta:

theta = asin(0.087 * lambda / d)

Now substitute the values:

theta = asin((0.087 * 590e-9 m) / (5.0e-3 m))

Calculating this using a calculator:

theta ≈ asin(1.02e-5) ≈ 0.000588 radians

To convert this to degrees, multiply by (180/pi):

theta ≈ 0.0337 degrees

Therefore, the angular width of the central diffraction maximum is approximately 0.0337 degrees.

Now, let's address your mistake in subtracting 0.087 degrees from the result. The value 0.087 in the formula represents the angle to the first minimum, not the full width of the central maximum. So, there is no need to subtract it from your result. The calculated value 0.0337 degrees represents the full angular width of the central maximum.

Hope this helps clarify the concept and correct your calculation.