An object is placed 15 cm from a convex mirror of curvature 90 cm. Calculate the image position and the magnification.

We are given:
• Object distance u=−15 cmu = -15 \, \text{cm} (The object distance is always negative for real objects).
• Curvature of the convex mirror R=90 cmR = 90 \, \text{cm}.
• We need to find the image position and the magnification.
Step 1: Use the mirror equation to find the image position
The mirror equation is:
1f=1v+1u\frac{1}{f} = \frac{1}{v} + \frac{1}{u}
where:
• ff is the focal length of the mirror,
• vv is the image distance,
• uu is the object distance.
For a mirror, the focal length is related to the curvature by:
f=R2f = \frac{R}{2}
Given that R=90 cmR = 90 \, \text{cm}, the focal length of the convex mirror is:
f=902=45 cmf = \frac{90}{2} = 45 \, \text{cm}
Now, substitute the values of ff and uu into the mirror equation:
145=1v+1−15\frac{1}{45} = \frac{1}{v} + \frac{1}{-15} 145=1v−115\frac{1}{45} = \frac{1}{v} - \frac{1}{15}
To solve for vv, first, add 115\frac{1}{15} to both sides:
1v=145+115\frac{1}{v} = \frac{1}{45} + \frac{1}{15}
Now, find a common denominator for the terms on the right-hand side:
1v=145+345=445\frac{1}{v} = \frac{1}{45} + \frac{3}{45} = \frac{4}{45}
Thus:
v=454=11.25 cmv = \frac{45}{4} = 11.25 \, \text{cm}
Since vv is positive, the image is formed on the same side as the reflecting surface of the convex mirror, and it is virtual.
Step 2: Find the magnification
The magnification mm of the mirror is given by the formula:
m=−vum = -\frac{v}{u}
Substituting the values of vv and uu:
m=−11.25−15=11.2515=0.75m = -\frac{11.25}{-15} = \frac{11.25}{15} = 0.75
Final Answer:
• The image is formed at a distance of 11.25 cm11.25 \, \text{cm} from the mirror (virtual image).
• The magnification is 0.750.75, meaning the image is smaller than the object and upright.