Optics Problems with Solutions: Mirrors, Lenses, Magnification, and Image Formation

Overview

This article gives you a compact study hub for common mirror and lens questions. The goal is not just to finish a few worked examples, but to build a repeatable method for solving image formation problems under exam pressure.

In introductory optics, most questions reduce to a small set of ideas:

• Identify the optical element: concave mirror, convex mirror, converging lens, or diverging lens.
• Assign the focal length and object distance using a consistent sign convention.
• Use the image equation: 1/f = 1/d_o + 1/d_i.
• Use the magnification equation: m = h_i/h_o = -d_i/d_o.
• Interpret the signs to decide whether the image is real or virtual, upright or inverted, enlarged or reduced.

A simple sign convention used in many school and college courses is:

• Converging lens: f positive
• Diverging lens: f negative
• Concave mirror: f positive
• Convex mirror: f negative
• Real image: d_i positive
• Virtual image: d_i negative
• Real object: d_o positive in standard introductory problems

Always confirm the sign convention your course uses. If your teacher or exam board defines distances differently, keep the formulas but adapt the signs consistently. If sign mistakes are common for you, it helps to keep a one-page reference nearby, much like an optics version of a physics equation sheet.

Before the worked examples, here are the principal rays most often used in ray diagram practice:

• A ray parallel to the principal axis reflects or refracts through the focal point for converging systems, or appears to come from the focal point for diverging systems.
• A ray through the focal point emerges parallel to the axis in converging systems.
• A ray through the center of a thin lens continues approximately straight.
• A ray through the center of curvature of a mirror reflects back on itself.

Ray diagrams are useful for visual reasoning, but on tests you should usually pair them with algebra. The diagram tells you what kind of answer makes sense. The equations tell you the exact values.

Worked example 1: Concave mirror outside the focal point

Problem. An object is placed 30 cm in front of a concave mirror with focal length 10 cm. Find the image distance and magnification. Describe the image.

Step 1: List known values.

• d_o = 30 cm
• f = +10 cm

Step 2: Use the mirror equation.

1/f = 1/d_o + 1/d_i

1/10 = 1/30 + 1/d_i

1/d_i = 1/10 - 1/30 = 2/30 = 1/15

So, d_i = 15 cm.

Step 3: Find magnification.

m = -d_i/d_o = -15/30 = -0.50

Interpretation.

• d_i is positive, so the image is real.
• m is negative, so the image is inverted.
• |m| is less than 1, so the image is smaller than the object.

Answer. The image forms 15 cm in front of the mirror, real, inverted, and half the object's height.

Worked example 2: Concave mirror inside the focal point

Problem. A 4 cm tall object is 6 cm in front of a concave mirror of focal length 12 cm. Find image distance and image height.

Step 1: Known values.

• d_o = 6 cm
• f = +12 cm
• h_o = 4 cm

Step 2: Mirror equation.

1/12 = 1/6 + 1/d_i

1/d_i = 1/12 - 1/6 = -1/12

d_i = -12 cm

Step 3: Magnification.

m = -d_i/d_o = -(-12)/6 = 2

Then:

h_i = m h_o = 2 × 4 = 8 cm

Interpretation.

• d_i negative means the image is virtual.
• m positive means the image is upright.
• |m| = 2 means the image is magnified.

Answer. The image is 12 cm behind the mirror, upright, virtual, and 8 cm tall.

Worked example 3: Convex mirror

Problem. An object is 20 cm in front of a convex mirror with focal length 8 cm in magnitude. Find the image distance and magnification.

For a convex mirror, use f = -8 cm.

Step 1: Equation setup.

1/f = 1/d_o + 1/d_i

1/(-8) = 1/20 + 1/d_i

1/d_i = -1/8 - 1/20 = -7/40

d_i = -40/7 ≈ -5.7 cm

Step 2: Magnification.

m = -d_i/d_o = -(-5.7)/20 ≈ 0.29

Interpretation.

• Negative d_i means virtual.
• Positive m means upright.
• Small |m| means reduced.

This is the standard pattern for convex mirrors: virtual, upright, and smaller images.

Worked example 4: Converging lens forming a real image

Problem. An object stands 24 cm from a converging lens of focal length 8 cm. Find the image location and magnification.

Step 1: Known values.

• d_o = 24 cm
• f = +8 cm

Step 2: Lens equation.

1/8 = 1/24 + 1/d_i

1/d_i = 1/8 - 1/24 = 2/24 = 1/12

d_i = 12 cm

Step 3: Magnification.

m = -12/24 = -0.50

Interpretation.

• Positive d_i: real image on the far side of the lens.
• Negative m: inverted.
• |m| = 0.50: image is half-size.

Worked example 5: Converging lens inside the focal length

Problem. A 2 cm high object is placed 10 cm from a converging lens with focal length 15 cm. Find image distance, magnification, and image height.

Step 1: Known values.

• d_o = 10 cm
• f = +15 cm
• h_o = 2 cm

Step 2: Lens equation.

1/15 = 1/10 + 1/d_i

1/d_i = 1/15 - 1/10 = -1/30

d_i = -30 cm

Step 3: Magnification and image height.

m = -(-30)/10 = 3

h_i = 3 × 2 = 6 cm

Interpretation.

• Negative d_i gives a virtual image.
• Positive m gives an upright image.
• |m| = 3 means the image is enlarged.

This is the familiar magnifying-glass case.

Worked example 6: Diverging lens

Problem. An object is 18 cm from a diverging lens with focal length 6 cm in magnitude. Find image distance and magnification.

For a diverging lens, f = -6 cm.

Step 1: Equation.

1/(-6) = 1/18 + 1/d_i

1/d_i = -1/6 - 1/18 = -4/18 = -2/9

d_i = -4.5 cm

Step 2: Magnification.

m = -(-4.5)/18 = 0.25

Interpretation.

Diverging lenses produce images that are typically virtual, upright, and reduced for real objects.

These six examples cover most image formation problems students see in school physics. If your difficulty is setting up the information from words alone, it is worth reviewing how to solve physics word problems step by step before doing more optics practice.

Maintenance cycle

Use this section as a regular refresh routine. Optics is not a topic you study once and keep forever. Sign conventions and ray patterns fade quickly if you do not revisit them.

A useful maintenance cycle is short and repeatable:

1. Weekly during an optics unit: solve two mirror problems and two lens problems without notes.
2. Before a quiz: redraw the four standard cases from memory: concave mirror, convex mirror, converging lens, diverging lens.
3. Before a midterm or final: build a one-page review sheet with formulas, sign rules, and one solved example of each image type.
4. After getting a problem wrong: classify the mistake as diagram, sign, algebra, or interpretation.

For each review cycle, do the same sequence:

• Sketch the object and optical element.
• Predict the image type before calculating.
• Calculate d_i.
• Calculate m and, if needed, h_i.
• Check whether your algebra matches your sketch.

This matters because optics problems are often lost on consistency, not on difficulty. A student may know the formulas but forget that a negative image distance usually means a virtual image. Another may solve correctly but report the image on the wrong side of the lens.

If you are revising multiple physics topics at once, pair this article with broader exam planning resources such as College Physics Midterm Study Guide: What to Review First, A-Level Physics Revision Checklist by Topic and Exam Season, or AP Physics 1 Formula Sheet Explained. Optics is easier to retain when it is reviewed in short bursts rather than in one long cram session.

Signals that require updates

This article is evergreen, but your own notes and problem set should still be updated from time to time. Here are the main signals that your optics review material needs a refresh.

• You keep mixing mirror and lens conventions. Add a side-by-side comparison table.
• You can calculate numbers but cannot describe the image. Include verbal summaries: real or virtual, upright or inverted, enlarged or reduced.
• Your course now emphasizes ray diagrams more heavily. Add sketch practice and label focal points and centers clearly.
• Your exam board uses a different sign convention. Rewrite your formula sheet to match that convention exactly.
• You are moving from basic to applied problems. Add questions involving image height, multiple representations, or experimental measurement.

Another signal is when search intent shifts in your own studying. Early in a unit, you may search for definitions and basic formulas. Closer to exams, what you really need is speed, worked examples, and mixed practice. At that stage, this topic becomes less about learning what a focal point is and more about recognizing common problem patterns fast.

If you are working from lab data rather than ideal textbook values, measurement quality matters too. Small errors in object distance can change your calculated image location. For that kind of review, see Measurement Uncertainty and Significant Figures in Physics Labs and Physics Unit Conversions Guide.

Common issues

This section addresses the mistakes that appear most often in optics homework help and exam prep.

1. Using the wrong sign for focal length

This is the most common issue. A converging lens and a concave mirror usually have positive focal length in introductory sign conventions. A diverging lens and a convex mirror usually have negative focal length. If every answer you get seems physically strange, check f first.

2. Forgetting what the sign of magnification means

Students often remember that magnification gives size change but forget that the sign also gives orientation.

• m positive: upright image
• m negative: inverted image

That sign is not an extra detail. It is part of the final interpretation.

3. Mixing up image distance with image height

d_i tells you where the image forms. h_i tells you how tall it is. The magnification equation connects them, but they are not interchangeable. Keep units and symbols clear.

4. Skipping the physical meaning of the answer

If a converging lens gives d_i negative for an object inside the focal length, that is not a mistake by itself. It means the image is virtual and on the same side as the object. The sign is the physics.

5. Drawing ray diagrams without checking scale or logic

A rough sketch is fine, but it should still reflect the correct case. If your algebra says the image is virtual and upright, but your ray diagram shows a real inverted image, at least one of them is wrong. Use the diagram as a reasonableness test.

6. Memorizing cases without understanding the pattern

There is a cleaner way to remember image behavior:

• Converging systems can make real or virtual images depending on object position.
• Diverging systems usually make virtual, upright, reduced images for real objects.

That pattern helps more than trying to memorize isolated outcomes.

7. Algebra slips in reciprocal equations

Optics equations look simple, but reciprocal arithmetic causes errors. Write each step carefully:

• Find a common denominator.
• Keep track of negative signs.
• Only invert after you have simplified correctly.

If algebra accuracy is a wider issue for you, practice with other step-based topics too. Graph interpretation and force analysis often improve the same habits of organized problem solving. Helpful companions include Graphing in Physics and the Free-Body Diagram Guide.

When to revisit

Return to this optics guide whenever you notice that the topic feels familiar but not automatic. That is the exact stage where short, high-quality review creates the most improvement.

Revisit this article:

• One week after first learning mirrors and lenses
• Before any unit quiz on geometric optics
• When starting cumulative exam prep
• After getting two or more image formation questions wrong in a row
• When switching between GCSE, A-Level, AP, or college-style question sets and needing to reset your sign convention

For a practical review session, use this 15-minute routine:

1. Write the two core equations from memory.
2. State the focal-length sign for each optical element.
3. Solve one mirror problem and one lens problem.
4. For each answer, label the image as real or virtual, upright or inverted, enlarged or reduced.
5. Sketch a quick ray diagram to verify your result.

If you have more time, build your own mini bank of optics problems with solutions. Include one example each for:

• Concave mirror with object beyond the focal point
• Concave mirror with object inside the focal point
• Convex mirror
• Converging lens with object beyond the focal point
• Converging lens with object inside the focal point
• Diverging lens

That set is compact enough to revisit regularly and broad enough to cover the patterns most exams test.

The real value of a solved-problems article is not just that it gives answers. It gives you a structure for your next round of practice. Use the examples here as templates, update your notes when your course emphasis shifts, and come back whenever optics problems stop feeling predictable. In physics exam prep, consistency usually beats volume, and geometric optics is a topic where a small, well-maintained review set goes a long way.