To solve the problem of finding the radius of curvature of a concave mirror where the object distance is 15 cm and the image distance is 5 cm, let's walk through the steps using the mirror formula and related concepts.
### Mirror Formula
The mirror formula is given by:
[tex]\[
\frac{1}{f} = \frac{1}{v} + \frac{1}{u}
\][/tex]
where:
- [tex]\( f \)[/tex] is the focal length of the mirror,
- [tex]\( v \)[/tex] is the image distance from the mirror,
- [tex]\( u \)[/tex] is the object distance from the mirror.
### Sign Conventions for Mirrors
For a concave mirror:
- The object distance [tex]\( u \)[/tex] is taken as negative since it is measured against the direction of the incident light (real object).
- The image distance [tex]\( v \)[/tex] is taken as positive if the image is real and negative if the image is virtual.
Given that the object distance [tex]\( u \)[/tex] is 15 cm (which we take as -15 cm following the sign convention for mirrors) and the image distance [tex]\( v \)[/tex] is 5 cm.
### Applying the Mirror Formula
Substitute the given values into the mirror formula:
[tex]\[
\frac{1}{f} = \frac{1}{v} + \frac{1}{u}
\][/tex]
Now substitute [tex]\( u = -15 \)[/tex] cm and [tex]\( v = 5 \)[/tex] cm:
[tex]\[
\frac{1}{f} = \frac{1}{5} + \frac{1}{-15}
\][/tex]
### Solving for the Focal Length [tex]\( f \)[/tex]
Calculate the right-hand side:
[tex]\[
\frac{1}{f} = \frac{1}{5} - \frac{1}{15}
\][/tex]
Find a common denominator:
[tex]\[
\frac{1}{5} = \frac{3}{15} \quad \text{and} \quad \frac{1}{-15} = -\frac{1}{15}
\][/tex]
Combine the fractions:
[tex]\[
\frac{1}{f} = \frac{3}{15} - \frac{1}{15} = \frac{2}{15}
\][/tex]
Thus, the focal length [tex]\( f \)[/tex] is:
[tex]\[
f = \frac{15}{2} = 7.5 \text{ cm}
\][/tex]
### Finding the Radius of Curvature [tex]\( R \)[/tex]
The radius of curvature [tex]\( R \)[/tex] of a concave mirror is related to the focal length by the formula:
[tex]\[
R = 2f
\][/tex]
Substitute the focal length:
[tex]\[
R = 2 \times 7.5 = 15 \text{ cm}
\][/tex]
Thus, the Radius of Curvature of the mirror is 15 cm.
