The radius of curvature of a concave mirror is 40 cm. An object is placed in front of it at a distance of 15 cm. Find the position of its image and draw its ray diagram.
Show Hint
Use the mirror formula \( \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \) and sign conventions carefully to locate images in concave mirrors.
Given:
Radius of curvature, \( R = 40 \, \text{cm} \)
Object distance, \( u = -15 \, \text{cm} \) (object is in front of mirror, so negative)
Step 1: Calculate the focal length \( f \):
\[
f = \frac{R}{2} = \frac{40}{2} = 20 \, \text{cm}
\]
Step 2: Use the mirror formula:
\[
\frac{1}{f} = \frac{1}{v} + \frac{1}{u}
\]
where \( v \) is the image distance.
\[
\frac{1}{v} = \frac{1}{f} - \frac{1}{u} = \frac{1}{20} - \frac{1}{-15} = \frac{1}{20} + \frac{1}{15} = \frac{3 + 4}{60} = \frac{7}{60}
\]
\[
v = \frac{60}{7} \approx 8.57 \, \text{cm}
\]
So, the image is formed at \( v = +8.57 \, \text{cm} \) in front of the mirror (real side).
Step 3: Nature of image: Since \( v \) is positive, the image is real and formed on the same side as the object. It is also smaller than the object since \( |v|<|u| \).
Ray diagram instructions (to be drawn by student):
Draw the principal axis and concave mirror with center of curvature \( C \) at 40 cm.
Mark the focal point \( F \) at 20 cm.
Place the object at 15 cm in front of the mirror.
Draw at least two rays:
A ray parallel to the principal axis, reflected through the focal point.
A ray passing through the focal point, reflected parallel to the principal axis.
The point where the reflected rays intersect in front of the mirror gives the position of the real image.
