What is f in mirror formula `(1)/(u)+(1)/(v)=(1)/(f)` ?

Options

A. distance of object
B. distance of image
C. focal length
D. None.

Answer

To find the value of \( f \) in the mirror formula \( \frac{1}{u} + \frac{1}{v} = \frac{1}{f} \), let's break it down step by step:

### Step 1: Understand the Mirror Formula
The mirror formula relates the object distance \( u \), the image distance \( v \), and the focal length \( f \) of a mirror. The formula is given as:
\[ \frac{1}{u} + \frac{1}{v} = \frac{1}{f} \]

### Step 2: Identify the Variables
- \( u \): Object distance from the mirror (measured from the mirror to the object).
- \( v \): Image distance from the mirror (measured from the mirror to the image).
- \( f \): Focal length of the mirror (the distance from the mirror's surface to its focal point).

### Step 3: Rearranging the Formula
To find \( f \), we can rearrange the mirror formula. Start by isolating \( \frac{1}{f} \):
\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]

### Step 4: Finding Focal Length \( f \)
Now, take the reciprocal of both sides to solve for \( f \):
\[ f = \frac{1}{\left(\frac{1}{u} + \frac{1}{v}\right)} \]

### Step 5: Sign Conventions
It is important to note the sign conventions used in optics. For mirrors:
- The object distance \( u \) is taken as negative if the object is in front of the mirror (real object).
- The image distance \( v \) is positive if the image is real and formed on the same side as the object.
- The focal length \( f \) is negative for concave mirrors and positive for convex mirrors.

### Conclusion
Thus, the focal length \( f \) in the mirror formula is derived as:
\[ f = \frac{uv}{u + v} \]
This gives us the focal length based on the object and image distances.

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