A concave mirror of focal length 20cm produces an erect image that is four times the object, the object distance from the mirror is

Analysis

We are given a concave mirror with a focal length ($f$) of 20 cm. The mirror produces an erect image that is four times the size of the object. We need to find the object distance ($u$).

Concept

1. Mirror Formula: $\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$
2. Magnification ($m$): $m = \frac{v}{u}$
3. Sign Convention for Concave Mirrors: • Focal length ($f$) is positive (+20 cm). • For a concave mirror to produce an erect image, the image must be virtual. • Virtual images are formed behind the mirror, so the image distance ($v$) is negative. • Magnification ($m$) for a virtual image is positive.

Solution

Step 1: Determine the relationship between $v$ and $u$. Given $m = 4$ and $m = \frac{v}{u}$: $4 = \frac{v}{u} \implies v = 4u$

Since the image is virtual (erect), we use $v = -4u$ in the mirror formula.

Step 2: Apply the mirror formula. $\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$ $\frac{1}{20} = \frac{1}{u} + \frac{1}{-4u}$

Step 3: Solve for $u$. Find a common denominator for the right side ($4u$): $\frac{1}{20} = \frac{4 - 1}{4u}$ $\frac{1}{20} = \frac{3}{4u}$

Cross-multiply: $4u = 60$ $u = \frac{60}{4}$ $u = 15 \text{ cm}$

Step 4: Compare with the options. We found the object distance to be 15 cm. Since 15 cm is less than the focal length of 20 cm, it falls into the category of "less than 20 cm."

Conclusion

The object must be placed between the pole and the principal focus of the concave mirror to produce an erect, magnified, virtual image. Since 15 cm < 20 cm, the correct option is A.

Exam Tip

For any concave mirror, remember this rule: if the object is placed at or beyond the focal point, the image is always real and inverted. The only way to get an erect (virtual) image with a concave mirror is to place the object between the pole and the focal point ($u < f$). Since your focal length is 20 cm, any answer that isn't less than 20 cm is physically impossible for an erect image!