Image Formation by a Concave Mirror (Candle, Screen & Movement)

We have a concave spherical mirror and a small candle placed in front of it.

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1. Given Data

• Radius of curvature: $R = 36\,\text{cm}$
• Object distance: $u = -27\,\text{cm}$ (by Cartesian sign convention, object in front of mirror is negative)
• Object (candle) height: $h_o = 2.5\,\text{cm}$

Using the relation between focal length and radius of curvature:

$$f = \frac{R}{2} = \frac{36}{2} = 18\,\text{cm}$$

So the focal length of the concave mirror is $f = 18\,\text{cm}$. For a concave mirror, $f < 0$ in strict sign-convention form. In the standard Cartesian form for mirrors:

• $f < 0$ for concave
• $u < 0$ for a real object in front
• Real image forms with $v < 0$

To keep the algebra and interpretation clear in this explanation, we’ll use the mirror formula in magnitude form first, then interpret signs.

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2. Mirror Formula

The spherical mirror formula:

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

Using magnitudes only (since everything lies in front of the mirror, we expect a real image):

• $f = 18\,\text{cm}$
• $u = 27\,\text{cm}$

$$\frac{1}{18} = \frac{1}{v} + \frac{1}{27}$$

Solve for $v$:

$$\frac{1}{v} = \frac{1}{18} - \frac{1}{27} = \frac{3 - 2}{54} = \frac{1}{54}$$

So,

$$v = 54\,\text{cm}$$

Distance of the screen from the mirror

A sharp image of the candle is formed 54 cm in front of the mirror, so the screen must be placed at:

$$\boxed{54\,\text{cm from the mirror}}$$

(Physically: on the same side as the object, at the real-image location.)

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3. Nature and Size of the Image

Nature

For a concave mirror with the object between $f$ and $2f$ (here $f = 18\,\text{cm}$, $2f = 36\,\text{cm}$, and $u = 27\,\text{cm}$), the image is:

• Real (can be obtained on a screen)
• Inverted
• Magnified (larger than the object)
• Formed beyond 2f (since $v = 54\,\text{cm} > 36\,\text{cm}$)

Linear magnification

Magnification $m$ for a mirror:

$$m = \frac{h_i}{h_o} = -\frac{v}{u}$$

Use magnitudes for the size; the sign only tells orientation.

$$m = \frac{v}{u} = \frac{54}{27} = 2$$

So the image is twice as tall as the object (but inverted):

$$h_i = m h_o = 2 \times 2.5\,\text{cm} = 5.0\,\text{cm}$$

Thus, the image is:

• $\boxed{\text{Real, inverted, and magnified}}$
• $\boxed{\text{Image height } = 5.0\,\text{cm}}$

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4. What Happens if We Move the Candle Closer to the Mirror?

Let the new object distance be $u' < 27\,\text{cm}$ (in magnitude: the candle is moved closer to the mirror).

Using the mirror formula again (in magnitudes):

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

with $f = 18\,\text{cm}$ fixed. If $u'$ decreases, then $1/u'$ increases, so to keep $1/f$ the same, $1/v'$ must decrease.

That means:

$$v' > v = 54\,\text{cm}$$

So as you move the candle closer to the mirror (still beyond the focal point):

• The image distance increases.
• The screen must be moved farther away from the mirror to remain at the new image location.

If the candle is brought exactly to the focal point ($u' = f = 18\,\text{cm}$), then:

$$\frac{1}{v'} = \frac{1}{f} - \frac{1}{u'} = 0 \Rightarrow v' \to \infty$$

The rays become parallel, and you cannot obtain a sharp image on a finite screen distance.

If the candle is moved inside the focal length (closer than 18 cm):

• $v'$ becomes negative in sign-convention form: the image is virtual, upright, and magnified, and it forms behind the mirror.
• No screen placed in front of the mirror will capture a sharp image; you must look into the mirror to see it.

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5. Summary

• Screen position for sharp image initially: $54\,\text{cm}$ from the mirror.
• Image nature: real, inverted, magnified.
• Image size: $5.0\,\text{cm}$ tall.
• If the candle is moved closer to the mirror (but still beyond the focal length): the image forms farther from the mirror, so the screen must be moved further away to maintain a sharp image.

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About the Visualization

The interactive visualization below lets you:

• Adjust the object distance to see how the real image and screen must move.
• Adjust the mirror curvature to change the focal length.
• See the principal rays and resulting image position and size.

Use it to explore what happens as the candle approaches the focal point and crosses inside it (where the image becomes virtual and no screen position works).