A rear mirror of a vehicle is cylindrical having radius of curvature 10 cm. The length of arc of curved surface is also 10 cm. If the eye of driver is assumed to be at large distance, from the mirror, then the field of view in radian is

Options

A. `2.0`
B. `4.0`
C. `3.0`
D. `5.0`

Answer

To find the field of view of a cylindrical rear mirror, we can follow these steps:

### Step 1: Understand the Geometry of the Mirror
The cylindrical mirror has a radius of curvature (R) of 10 cm. The length of the arc of the curved surface is also given as 10 cm.

### Step 2: Determine the Focus of the Mirror
For a cylindrical mirror, the focal length (f) is given by the formula:
\[ f = \frac{R}{2} \]
Substituting the radius of curvature:
\[ f = \frac{10 \, \text{cm}}{2} = 5 \, \text{cm} \]

### Step 3: Calculate the Field of View
The field of view (θ) can be calculated using the formula:
\[ \theta = \frac{\text{Length of arc}}{\text{Radius of curvature}} \]
Here, the length of the arc is 10 cm and the radius of curvature can be considered as the distance from the focus to the edge of the arc, which is equal to the focal length (5 cm):
\[ \theta = \frac{10 \, \text{cm}}{5 \, \text{cm}} = 2 \, \text{radians} \]

### Step 4: Conclusion
Thus, the field of view of the cylindrical rear mirror is:
\[ \theta = 2 \, \text{radians} \]

### Final Answer
The field of view in radians is **2 radians**.

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