E. Physics: Light and Geometric Optics · Light and Geometric Optics

Ray Diagrams for Concave Mirrors

E2.3 E3.3 A1.5 A1.8 A1.10About 75 minutes

Learning goals

I can draw accurate ray diagrams for concave mirrors using at least two special rays.
I can predict the size, attitude, location and type of the image for any object position.
I can explain why an object at F produces no image and why an object inside F produces a virtual image.
I can test my predictions with a concave mirror and compare predicted and observed results.

Words to know

converging mirror
Another name for a concave mirror.

inverted
Upside down compared with the object.

upright
The same way up as the object.

real image
An image formed where reflected rays actually cross; it can be projected on a screen.

virtual image
An image formed where extended rays meet behind the mirror; it cannot be projected.

scale diagram
A drawing in which all lengths are in the same proportion to the real lengths.

Upside down far away, right side up up close

When you looked into the inside of a spoon, your image was upside down from far away and right side up when you held the spoon very close. Concave mirrors can make many different kinds of images, depending on where the object is. Today you will learn to predict the image with a ray diagram, and then test your predictions.

How to draw a concave-mirror ray diagram

Draw the principal axis and the mirror (a gentle curve through V). Mark F and C to scale (C is at 2f).
Draw the object as an upright arrow standing on the principal axis, at the right distance from the mirror.
From the tip of the object, draw at least two of the special rays:

Ray 1: parallel to the principal axis → reflects through F.
Ray 2: through F → reflects parallel to the axis.
Ray 3: through C → reflects back on itself.

Where the reflected rays cross, that is the tip of the image. Draw the image arrow from the principal axis to that point.
If the reflected rays spread apart and never cross in front of the mirror, extend them backwards behind the mirror with dashed lines. Where the dashed lines cross is the tip of a virtual image.
Describe the image using SALT.

Object beyond C. The three reflected rays cross between C and F. The image is smaller, inverted, between C and F, and real (light really passes through it, so it could be caught on a screen).

Worked example

Example 1: Let's draw one to scale

A concave mirror has a focal length of 4.0 cm. A 2.0 cm tall object stands 6.0 cm in front of it. Draw a scale ray diagram and describe the image.

Scale: draw lengths full size (1 cm on paper = 1 cm). Mark F at 4.0 cm and C at 8.0 cm from V. The object is at 6.0 cm: between C and F.
Ray 1: from the object's tip, parallel to the axis to the mirror, then reflected through F and beyond.
Ray 2: from the tip through F to the mirror (it hits below the axis), then reflected parallel to the axis.
Ray 3: from the tip along the line from C, reflected straight back through C.
The three reflected rays cross 12 cm from the mirror, 4.0 cm below the axis. Draw the image arrow pointing down from the axis to that point.
SALT: S – larger (4.0 cm vs 2.0 cm); A – inverted; L – 12 cm in front of the mirror, beyond C; T – real.

The image is larger (4.0 cm tall), inverted, 12 cm in front of the mirror (beyond C), and real.

Example 1 drawn to scale (f = 4.0 cm, object 6.0 cm away and 2.0 cm tall). The rays cross beyond C, making a larger, inverted, real image twice as tall as the object.

When there is no image

If the object is placed exactly at F, every reflected ray leaves parallel to every other one. Parallel rays never meet, and their backward extensions never meet either, so no image forms. This is not a failure: it is exactly how a flashlight, car headlight or lighthouse makes a beam. The bulb sits at F, and the light leaves as a strong parallel beam.

Object at F. Ray 1 reflects through F; ray 3 reflects back through C. The two reflected rays are parallel, so they never meet: no image.

Inside the focal point: a virtual image

When the object is between F and the mirror, the reflected rays diverge (spread apart) in front of the mirror. Your eye traces them back, and the dashed extensions meet behind the mirror. The image is larger, upright, behind the mirror, and virtual. This is the image you see in a magnifying makeup or shaving mirror, or in a dentist's mirror held close to a tooth.

For ray 2, the object is between F and the mirror, so the ray cannot pass through F on its way to the mirror. Instead, draw it along the line from F through the object's tip (as if it came from F); it still reflects parallel to the axis.

Object between F and V. The reflected rays spread apart; their dashed extensions meet behind the mirror. The image is larger, upright and virtual (here about 2.7 times taller than the object).

Object position	Image location	Size	Attitude	Type	Example use
very far away	at F	much smaller	inverted	real	reflecting telescope, solar cooker
beyond C	between C and F	smaller	inverted	real	telescope mirrors (distant objects)
at C	at C	same size	inverted	real	measuring R and f in the lab
between C and F	beyond C	larger	inverted	real	making an enlarged image on a screen
at F	no image	—	—	—	flashlights, headlights, lighthouses
between F and V	behind the mirror	larger	upright	virtual	makeup and shaving mirrors, dentist's mirror

Summary for a concave mirror. Notice the pattern: as the object moves in from far away towards F, the real image moves out from F and gets bigger.

Checks that catch mistakes

The image of an object beyond F is always real and inverted; inside F it is always virtual and upright.
A real image is always in front of the mirror (on the same side as the object); a virtual image is behind it.
If you draw three rays and they do not cross at one point, re-check your rays: one of them was probably drawn through the wrong point.
Use a ruler; small errors near F grow large in the image.

Just for fun

A concave mirror starts every modern Olympic torch relay: at Olympia, Greece, the flame is lit by sunlight focused by a curved mirror. That's what we call a hot focal point.

Hands-on activity

Test your predictions with a makeup mirror

You need: a concave makeup or shaving mirror (found f in the last lesson), a small bright object such as a phone screen showing a bold arrow or a small battery-powered LED light, a sheet of white card as a screen, a metre stick or measuring tape, a dim room.

Using your value of f, mark distances of 2f (C) and f (F) from the mirror on the table with tape.
Predict (with a quick ray diagram) the image for the object at: about 3f, at 2f, between 2f and f, and between f and the mirror.
Test. Place the object at each position. For the first three, move the card screen until a sharp image appears on it. Record the image distance, size compared with the object, and whether it is upright or inverted.
For the last position, try to catch an image on the screen (you can't). Look into the mirror instead and describe what you see.
Compare your predictions with your observations. Where did they disagree, and why might that be?

What to look for: at 3f a small, inverted image just beyond F; at 2f an inverted image about the same size at 2f; between 2f and f a large, inverted image far from the mirror (you may need to move the screen well back); inside f, no image on the screen but a large, upright image in the mirror.

Safety (grown-up note): Do this indoors, away from sunlight; a concave mirror can focus sunlight hot enough to start a fire.

Your turn

Name the three special rays used for concave-mirror ray diagrams.

An object is beyond C. Describe the image using SALT.

An object is between F and the mirror. Describe the image using SALT.

Where should a bulb be placed in front of a concave mirror to produce a parallel beam? Explain.

How can you tell from a ray diagram whether an image is real or virtual?

A concave mirror has f = 10 cm. An object is placed 20 cm away. Predict the image.

More practice: warm-up

What is the minimum number of rays needed to locate an image?

Is a real image formed in front of or behind a concave mirror?

Which object position gives no image?

Which object position gives an image the same size as the object?

What kind of mirror is a magnifying makeup mirror?

More practice: on your own

Explain why the image of an object beyond C is smaller than the object.

A dentist holds a small concave mirror 1.5 cm from a tooth. The mirror's focal length is 2.5 cm. Describe the image the dentist sees and explain why this is useful.

As an object moves from far away towards F, what happens to the location and size of its real image?

In a ray diagram, a student's ray 2 for an object inside F was drawn from the tip through F and then to the mirror. What is wrong, and how should it be drawn?

Use a ray diagram (or the table) to predict the image when a concave mirror with f = 15 cm has an object at 25 cm.

Challenge

A concave mirror forms a real image the same size as the object, 48 cm from the mirror. Find f and R.

Explain, using the reversibility of light, why an object between C and F gives an image beyond C, if you already know that an object beyond C gives an image between C and F.

A student says: "A concave mirror can never make an upright image." Give a counterexample and explain.

Check your understanding

Quiz

A concave mirror forms an image of an object placed beyond C. The image is:

Where should an object be placed to get a larger, upright, virtual image in a concave mirror?

For a concave mirror, a ray through F reflects:

An object is placed exactly at F of a concave mirror. The reflected rays:

A real image formed by a concave mirror is always:

A concave mirror has f = 12 cm. An object 24 cm away forms an image:

For parents and teachers

Big idea: Expectation E2.3 asks students to predict image characteristics (size, attitude, location, type) using ray diagrams and to test their predictions. The pattern in the summary table is worth understanding rather than memorizing: as the object approaches F, the real image moves outward and grows; inside F the image becomes virtual and upright.

Common difficulties:

Rays through the wrong point. Ask: "Which rule is this ray? Does it go through F or come out parallel?"
Ray 2 when the object is inside F. The ray is drawn along the line from F through the tip, not through F first.
Calling every upright image virtual without checking. For single curved mirrors this is true, and it is a useful check.

How to help: A magnifying makeup mirror and a phone flashlight make a great at-home lab. Ask your teen to predict, then test. Keep concave mirrors out of the Sun.