Solution: Concave Mirror Optics

Step 1 (Formula / Governing Law & Given Data): [0.5 Mark]

Given data based on sign convention:

• Object distance, $u = -30 \text{ cm}$
• Radius of curvature, $R = -40 \text{ cm}$
• Focal length, $f = \frac{R}{2} = \frac{-40}{2} = -20 \text{ cm}$

The mirror formula is given by:

$$\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$$

Step 2 (Step-by-Step Mathematical Working): [1.0 Mark]

(i) Position of the image ($v$):

$$\frac{1}{v} = \frac{1}{f} - \frac{1}{u}$$ $$\frac{1}{v} = \frac{1}{-20} - \frac{1}{-30}$$ $$\frac{1}{v} = -\frac{1}{20} + \frac{1}{30} = \frac{-3 + 2}{60} = -\frac{1}{60}$$ $$v = -60 \text{ cm}$$

(ii) Magnification ($m$):

$$m = -\frac{v}{u}$$ $$m = -\left( \frac{-60}{-30} \right) = -2$$

Step 3 (Final Concluded Answer): [0.5 Mark]

• (i) Position: The image is formed at $60 \text{ cm}$ in front of the mirror (real image).
• (ii) Magnification: The magnification is $-2$ (the image is inverted and twice the size of the object).

💡 EL10 Concept Insight:

Think of the concave mirror like a curved spoon; because the object is placed between the focal point and the center of curvature, the light rays bounce off and meet further away, creating a larger, upside-down version of the object on the same side.