Images Formed by Mirrors: Study Guide (Physics 2 13.2)

Curved mirrors and the focal point

In Topic 13.1 every reflection followed θᵢ = θᵣ. A curved mirror obeys the same law at every point, but the normal changes direction across the surface. That is what lets a curved mirror bring light together or spread it out.

This course uses three kinds of mirror: plane, concave spherical and convex spherical. A spherical mirror is a piece of the surface of a sphere. Its centre of curvature C is the centre of that sphere, and its radius of curvature R is the sphere’s radius. The principal axis is the line through C and the centre of the mirror. The point where the axis meets the mirror is the vertex.

• Concave (converging) mirror. The reflecting side curves inward, like the inside of a spoon. Rays arriving parallel to the principal axis reflect and pass through one point on the axis, the focal point F, in front of the mirror.
• Convex (diverging) mirror. The reflecting side bulges outward. Rays parallel to the axis reflect and spread out as if they came from one point behind the mirror. That point is its focal point.
• Plane mirror. Parallel rays stay parallel after reflection, so they never meet. The focal point is infinitely far away.

The focal length f is the distance from the vertex to F. For a spherical mirror, F lies about halfway between the mirror and C:

f ≈ R/2

This holds well for rays close to the axis and mirrors that are small compared with R, which is what this course assumes. In ray diagrams it is usual to draw each reflection at a vertical line through the vertex, as in the figures below.

Real and virtual images

Light leaves every point of an object in many directions. An image of that point forms where the reflected rays from it meet.

• A real image forms when reflected rays from one object point actually meet at one point. Light really passes through the image, so you can catch a real image on a screen. Only a concave mirror (with the object outside F) gives a real image.
• A virtual image forms when the reflected rays diverge, so that they only appear to come from one point behind the mirror. No light passes through that point, so a screen there shows nothing. Your eye can still see the image by collecting the diverging rays.

In diagrams, draw the real paths of light as solid lines. Draw the backward extensions of reflected rays, which light does not follow, as dashed lines.

The three principal rays

You can draw any number of rays from the top of an object. Three are easy to draw because the law of reflection makes their reflected paths predictable:

1. Parallel ray. A ray parallel to the principal axis reflects through F (concave), or reflects so that it appears to come from F (convex).
2. Vertex ray. A ray that strikes the vertex reflects symmetrically about the axis, like a ray hitting a plane mirror. The angle above the axis on the way in equals the angle below it on the way out.
3. Focal ray. A ray that passes through F (concave), or is heading towards F behind the mirror (convex), reflects parallel to the axis. This is ray 1 reversed.

Two rays are enough to locate the top of the image; the third is a check. The bottom of an object on the axis images onto the axis.

The mirror equation and sign conventions

The image position depends on the focal length and on the object distance. Ray geometry gives the mirror equation:

1/sᵢ + 1/sₒ = 1/f

where sₒ is the object distance and sᵢ the image distance, both measured from the vertex. You need a sign convention so that the equation tells you which side of the mirror things are on. This course uses the standard one:

Quantity	Positive (+)	Negative (−)
sₒ	object in front of the mirror (always, here)	—
sᵢ	image in front of the mirror: real	image behind the mirror: virtual
f	concave mirror	convex mirror

For a plane mirror f is infinite, so 1/f = 0 and sᵢ = −sₒ: the image is as far behind the mirror as the object is in front.

Magnification

The magnification compares the size of the image with the size of the object, and it depends on where the object and image are:

|M| = |hᵢ/hₒ| = |sᵢ/sₒ|

This is the form the course uses, with absolute-value bars on every part. It gives the size ratio only, so it never carries a sign. |M| > 1 means enlarged, |M| < 1 reduced, |M| = 1 same size. Decide the orientation from the type of image: for a single mirror, a real image is inverted and a virtual image is upright. (Some textbooks drop the bars and write M = −sᵢ/sₒ, so that a negative M signals an inverted image. That gives the same description, but in this course quote the magnitude and state the orientation in words.)

A full description of an image states four things: location, type (real or virtual), orientation (upright or inverted) and size (enlarged, reduced or same size).

Mirror and object position	Image location	Type and orientation	Size
Concave, object beyond C	between F and C	real, inverted	reduced
Concave, object at C	at C	real, inverted	same size
Concave, object between C and F	beyond C	real, inverted	enlarged
Concave, object at F	no image (reflected rays parallel)	—	—
Concave, object inside F	behind mirror	virtual, upright	enlarged
Convex, any position	behind mirror, between vertex and F	virtual, upright	reduced
Plane, any position	same distance behind	virtual, upright	same size

Do not memorise the table blindly: each row comes from a quick ray diagram or from the mirror equation.

Worked example 1: concave mirror, object beyond C

Question. A 4.0 cm tall object stands on the axis 30 cm in front of a concave mirror of radius of curvature 24 cm (Figure 1). Locate and describe the image.

1. Focal length: f = R/2 = 24 cm ÷ 2 = 12 cm, positive because the mirror is concave.
2. Mirror equation: 1/sᵢ = 1/f − 1/sₒ = 1/12 − 1/30 = 5/60 − 2/60 = 3/60, so sᵢ = +20 cm.
3. sᵢ is positive, so the image is real and 20 cm in front of the mirror. It is between F (12 cm) and C (24 cm), as the table predicts.
4. Magnification: |M| = |sᵢ/sₒ| = 20/30 = 0.667. Image height hᵢ = 0.667 × 4.0 cm = 2.7 cm.

Answer. Real, inverted, reduced image, 2.7 cm tall, 20 cm in front of the mirror.

Check. In Figure 1 the three principal rays meet 20 cm from the mirror, below the axis, at a point 2.7 cm from it. The calculation and the drawing agree.

Worked example 2: a convex security mirror

Question. A shop uses a convex security mirror of focal length 0.40 m. A customer 1.5 m tall stands 6.0 m in front of it. Locate the image and find its height.

1. Convex mirror, so f = −0.40 m.
2. 1/sᵢ = 1/f − 1/sₒ = 1/(−0.40) − 1/6.0 = −2.5 − 0.1667 = −2.667 m⁻¹, so sᵢ = −0.375 m.
3. sᵢ is negative: the image is virtual, 0.375 m behind the mirror, and therefore upright.
4. |M| = |sᵢ/sₒ| = 0.375/6.0 = 0.0625, so hᵢ = 0.0625 × 1.5 m = 0.094 m (9.4 cm).

Answer. Virtual, upright image 0.375 m (about 0.38 m) behind the mirror, 9.4 cm tall.

Interpretation. The image is small and close to F, whatever the customer’s distance. That is why convex mirrors show a wide field of view: a whole aisle fits in a small image. A plane mirror would give a 1.5 m image 6.0 m behind it, so it would show far less of the shop in the same area of glass.

Worked example 3: deriving where to stand (symbolic)

Question. A concave make-up mirror has focal length f. Where must a face be placed to give an upright image twice the size of the face? Evaluate for f = 18 cm.

1. Upright with one mirror means virtual, so sᵢ is negative. |M| = 2 means |sᵢ| = 2sₒ, so sᵢ = −2sₒ.
2. Substitute into the mirror equation: 1/(−2sₒ) + 1/sₒ = 1/f.
3. Simplify the left side: −1/(2sₒ) + 2/(2sₒ) = 1/(2sₒ). So 1/(2sₒ) = 1/f, which gives sₒ = f/2.
4. For f = 18 cm: sₒ = 9.0 cm, and sᵢ = −18 cm.

Check. sₒ = f/2 is inside F, which is the only region where a concave mirror gives an upright, enlarged image. Substitute back: 1/(−18) + 1/9 = 1/18 ✓.

Common misconceptions

• “The image is on the surface of the mirror.” A plane-mirror image is as far behind the mirror as the object is in front. Your reflection in a mirror 2 m away is 4 m from you.
• “A virtual image cannot be seen.” You see it whenever you look in a bathroom mirror. It cannot be projected on a screen.
• “A mirror swaps left and right.” A plane mirror reverses front and back (the direction perpendicular to the glass). Left stays left and up stays up; the reversed writing comes from that front-back reversal.
• “Step back to see more of yourself in a plane mirror.” A plane mirror needs to be at least half your height to show your whole body, and this does not depend on how far away you stand.
• Using R in place of f. Halve the radius of curvature first.
• Forgetting the minus sign for a convex mirror. With f positive, a convex mirror would wrongly appear to make a real image.
• “An object at F makes an image at F.” The reflected rays leave parallel, so no image forms at any finite distance.
• “Covering half of a concave mirror removes half the image.” Every part of the mirror receives light from every point of the object, so the whole image remains, only dimmer.

Where this leads

Topic 13.3 introduces refraction, and Topic 13.4 uses the same equation, principal rays and sign ideas for lenses: see Refraction. Now test yourself with the practice questions, then use the revision notes and the topic checklist.