Distance until a spherical wave is seen as a plane wave.

[Bhumble]

Homework Statement

A point source emits a spherical wave with lambda = 450 nm.
If an observer is far away from the source and he/she is only interacting with the light across a small area, one can approximate the local wave as a plane wave. How far from the source must the observer be so that the phase of the wave deviates from that of a plane wave by less than 45° (pi/4 rad) over an illuminated spot 2 cm in diameter?

Homework Equations

The Attempt at a Solution

I'm not even sure if my equations are correct but this is what I've gathered/assumed so far.
Spherical Wave

$$\Psi = \frac{A}{r} \cos{r +/- vt}$$

Plane Wave

$$\Psi = A \cos{x -/+ vt}$$

I'm set them equal to each other and solved for x which left me with x= r -/+ 2ct.
Since this is light.

Any help with problem setup is appreciated.

 

[mark726]

Hi there,

You are on the right track with your equations. However, there are a few things to consider in order to properly set up this problem.

First, make sure you are using the correct equation for a spherical wave. The equation you have written is for a plane wave, not a spherical wave. The correct equation for a spherical wave is:

\Psi = \frac{A}{r} e^{ikr}

where A is the amplitude, r is the distance from the source, k is the wave number (equal to 2\pi/\lambda), and e is the base of the natural logarithm.

Next, you will need to consider the geometry of the problem. The observer is far away from the source, so the wave can be approximated as a plane wave. This means that the wavefronts are nearly parallel, and the wave can be described by a single value for the phase (x in your equation). This value will be the same at any point along the wavefront.

To find the distance at which the phase deviates by less than 45° over a 2 cm diameter spot, you will need to use the equation for a circle:

(x - x_0)^2 + (y - y_0)^2 = r^2

where (x_0, y_0) is the center of the spot and r is the radius (1 cm in this case). This equation will give you a range of values for x, representing the points within the spot where the phase deviates by less than 45°.

Finally, you can use the equation for a plane wave to find the distance from the source at which the phase deviates by 45°. This will give you two possible values, one for each direction of propagation (i.e. -/+ in your equation).

I hope this helps. Let me know if you have any further questions.