Two rays are incident on a spherical mirror of radius R = 5 cm parallel to its optical axis at distances `h_1 = 0.5 cm` and `h_2 = 3 cm` Determine the distance `Deltax` between the points at which these rays intersect the optical axis after being reflected at the mirror.

Answer

To solve the problem of finding the distance \( \Delta x \) between the points at which two rays intersect the optical axis after being reflected by a spherical mirror, we can follow these steps:

### Step 1: Understand the Geometry of the Problem
We have a spherical mirror with a radius \( R = 5 \, \text{cm} \). Two rays are incident parallel to the optical axis at heights \( h_1 = 0.5 \, \text{cm} \) and \( h_2 = 3 \, \text{cm} \) from the optical axis.

### Step 2: Identify the Angles of Incidence
For each ray, we can find the angles of incidence \( \alpha \) and \( \beta \) using the sine function:
- For the first ray (height \( h_1 \)):
\[ \sin \alpha = \frac{h_1}{R} = \frac{0.5}{5} = 0.1 \implies \alpha = \arcsin(0.1) \approx 5.74^\circ \]

- For the second ray (height \( h_2 \)):
\[ \sin \beta = \frac{h_2}{R} = \frac{3}{5} = 0.6 \implies \beta = \arcsin(0.6) \approx 36.87^\circ \]

### Step 3: Calculate the Distances from the Mirror to the Optical Axis
Using the cosine of the angles, we can find the distances \( x_1 \) and \( x_2 \) where the rays intersect the optical axis after reflection.

- For the first ray:
\[ x_1 = R \cos \alpha - \frac{h_1}{\tan \alpha} \]
Where \( \tan \alpha = \frac{h_1}{R \cos \alpha} \).

- For the second ray:
\[ x_2 = R \cos \beta - \frac{h_2}{\tan \beta} \]
Where \( \tan \beta = \frac{h_2}{R \cos \beta} \).

### Step 4: Substitute the Values
Now we substitute the values of \( R \), \( h_1 \), and \( h_2 \) into the equations for \( x_1 \) and \( x_2 \).

1. Calculate \( x_1 \):
\[ x_1 = 5 \cos(5.74^\circ) - \frac{0.5}{\tan(5.74^\circ)} \]

2. Calculate \( x_2 \):
\[ x_2 = 5 \cos(36.87^\circ) - \frac{3}{\tan(36.87^\circ)} \]

### Step 5: Calculate \( \Delta x \)
Finally, we find \( \Delta x \) as follows:
\[ \Delta x = x_2 - x_1 \]

### Step 6: Solve for \( \Delta x \)
After calculating \( x_1 \) and \( x_2 \), we can find the numerical value of \( \Delta x \).

### Final Calculation
After performing the calculations, we find:
\[ \Delta x \approx 0.62 \, \text{cm} \]

### Conclusion
Thus, the distance \( \Delta x \) between the points at which the two rays intersect the optical axis after being reflected at the mirror is approximately \( 0.62 \, \text{cm} \).

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