13.4 Images Formed by Lenses

Lenses form images by refracting light. Convex (converging) lenses can make real or virtual images depending on where the object sits, while concave (diverging) lenses always make smaller, upright virtual images. In AP Physics 2, this topic connects ray diagrams, focal length, image distance, and magnification in geometric optics problems.

Why This Matters for the AP Physics 2 Exam

Geometric optics makes up a noticeable slice of the AP Physics 2 exam, and lenses are a core part of it. This topic gives you two skills the exam rewards: building accurate ray diagrams and using the thin-lens and magnification equations together. You need to do both because a calculation tells you distances and signs, while a ray diagram lets you confirm whether an image is real or virtual and upright or inverted.

Lenses also fit the experimental side of the course. Plotting image distance against object distance and linearizing the data is a natural fit for the kind of data analysis and experimental reasoning the exam asks you to do, so being comfortable with these relationships helps you explain procedures and interpret graphs.

Key Takeaways

• A convex lens converges parallel rays to a focal point on the far side; a concave lens spreads rays so they seem to come from a focal point on the near side.
• Use the thin-lens equation $\frac{1}{s_o}+\frac{1}{s_i}=\frac{1}{f}$ with consistent sign conventions: convex lenses have positive $f$, concave lenses have negative $f$.
• Real images come from rays actually meeting (positive $s_i$) and are inverted; virtual images come from rays only appearing to meet (negative $s_i$) and are upright.
• Magnification $M=-\frac{s_i}{s_o}=\frac{h_i}{h_o}$ tells you size and orientation. Negative $M$ means inverted, $|M|>1$ means enlarged.
• Three principal rays locate any image: parallel-to-axis, through the lens center, and through (or toward) a focal point.
• Convex lenses make real images when the object is beyond the focal point and virtual images when the object is inside the focal length; concave lenses only make virtual images.

Image Formation by Lenses

Convex Lens Refraction

Convex lenses bulge outward in the middle and cause light rays to converge. When parallel light rays strike a convex lens, they bend inward and meet at a single point on the opposite side called the focal point.

• These lenses are also known as converging lenses because they cause light to come together
• The distance from the lens to the focal point is called the focal length
• Convex lenses have positive focal lengths in optical calculations
• Common examples include magnifying glasses, camera lenses, and telescope lenses

A magnifying glass shows this when it focuses sunlight to a small, intense point. The lens concentrates the sun's nearly parallel rays to a single focal point.

Concave Lens Refraction

Concave lenses curve inward in the middle and cause light rays to spread apart. When parallel light rays pass through a concave lens, they diverge and appear to originate from a focal point on the same side that the light entered.

• These are called diverging lenses since they make light rays move away from each other
• The focal point of a concave lens is virtual, since light does not actually pass through it
• Concave lenses have negative focal lengths in optical calculations
• They are commonly used in eyeglasses to correct nearsightedness by spreading out light before it reaches the eye

When you look through a concave lens at an object, it always appears smaller than it actually is, showing the diverging effect on light rays.

Real Image Formation

Real images form when light rays actually converge at a location after passing through a lens.

• Light physically passes through the location where the image forms
• Real images can be projected onto a screen because light rays actually meet there
• They are typically inverted compared to the object
• Only convex lenses can form real images, and only when the object is beyond the focal point
• Examples include the image on a movie screen or on a camera's sensor

When a projector displays an image on a screen, you are seeing a real image formed by the projector's convex lens system.

Virtual Image Formation

Virtual images occur when light rays appear to diverge from a point but do not actually pass through that point. Your eye interprets these diverging rays as coming from an apparent location.

• Light rays do not physically meet at the image location but appear to originate from there
• Cannot be captured on a screen since no light actually passes through the image location
• Usually appear upright
• Both concave and convex lenses can form virtual images under certain conditions

When you use a magnifying glass to examine something closely, with the object inside the focal length, you are looking at a virtual image that appears larger and sits on the same side of the lens as the object.

Thin-Lens Equation

The thin-lens equation relates the object distance, image distance, and focal length. It works for both convex and concave lenses.

$\frac{1}{s_o}+\frac{1}{s_i}=\frac{1}{f}$

Where:

• $s_o$ is the object distance (distance from object to lens)
• $s_i$ is the image distance (distance from lens to image)
• $f$ is the focal length of the lens

This equation lets you calculate any one value if you know the other two. For example, if an object is 20 cm from a lens with a focal length of 10 cm:

$\frac{1}{20}+\frac{1}{s_i}=\frac{1}{10}$ $\frac{1}{s_i}=\frac{1}{10}-\frac{1}{20}=\frac{2-1}{20}=\frac{1}{20}$ $s_i=20 \text{ cm}$

Lens Focal Point Conventions

Sign conventions matter for correctly applying the thin-lens equation and reading your results.

A thin lens has two focal points, one on each side along the principal axis. For a converging (convex) lens, rays parallel to the axis from either side are brought to a focal point on the opposite side, so each side has a corresponding focal point. For a diverging (concave) lens, rays parallel to the axis spread out as if coming from a focal point on the incident side. The lens still has a focal point on each side, set by the lens shape and the direction the light comes from.

The sign conventions for the thin-lens equation:

• For convex lenses, the focal length is positive; for concave lenses, it is negative
• Object distances ($s_o$) are positive when the object is on the side where light originates
• Image distances ($s_i$) are positive when the image forms on the opposite side from where light enters
• Negative image distances indicate virtual images (on the same side as the incoming light)
• Keep these conventions consistent every time you solve

For example, a concave lens with a focal length of -15 cm always produces a virtual image, shown by a negative value for $s_i$ when you solve.

Image Magnification

Magnification tells you how much larger or smaller the image is than the object, and whether it is upright or inverted.

For a thin lens, the magnitude of the magnification is:

$|M|=\left|\frac{h_i}{h_o}\right|=\left|\frac{s_i}{s_o}\right|$

In the sign convention used with the thin-lens equation, the signed magnification is:

$M=-\frac{s_i}{s_o}=\frac{h_i}{h_o}$

Where:

• $M$ is the magnification
• $h_i$ is the image height
• $h_o$ is the object height
• $s_i$ is the image distance
• $s_o$ is the object distance

A positive $M$ means the image is upright; a negative $M$ means it is inverted. If $|M| > 1$, the image is enlarged; if $|M| < 1$, it is reduced; if $|M| = 1$, it is the same size as the object.

For instance, if $M = -2$, the image is inverted and twice as large. If $M = 0.5$, the image is upright and half the size.

Ray Diagrams for Lenses

Ray diagrams let you visualize how a lens forms an image. By tracing specific rays, you can find the image's location, size, orientation, and type.

Trace three key rays from the top of the object:

1. A ray parallel to the principal axis. After the lens, this ray:

   • For a convex lens: passes through the far-side focal point
   • For a concave lens: appears to come from the near-side focal point

2. A ray through the center of the lens. This ray continues straight through without bending.

3. A ray through (or toward) a focal point. After the lens, this ray:

   • For a convex lens: a ray directed through the near-side focal point emerges parallel to the principal axis
   • For a concave lens: a ray aimed toward the far-side focal point emerges parallel to the principal axis

Where these rays intersect (or appear to intersect when extended backward), they form the image. The diagram immediately shows:

• Whether the image is real (rays actually meet) or virtual (rays only appear to meet when extended backward)
• Whether the image is upright or inverted
• The relative size compared to the object

Ray diagrams are great for quickly predicting image characteristics, though they are less precise than the thin-lens equation.

How to Use This on the AP Physics 2 Exam

Problem Solving

• Start by listing $s_o$, $f$, and their signs before plugging into the thin-lens equation. A wrong sign on $f$ is the most common source of error.
• Solve for $\frac{1}{s_i}$ first, then take the reciprocal. Do not forget the final reciprocal step.
• Check your answer's sign: positive $s_i$ means a real image on the far side, negative $s_i$ means a virtual image on the near side.
• Use $M=-\frac{s_i}{s_o}$ to finish the description with orientation and size.

Ray Diagrams

• Draw at least two of the three principal rays. The third is a useful check.
• For a convex lens with the object inside the focal length, the refracted rays diverge on the far side, so extend them backward to the object side to locate the upright, enlarged virtual image.
• Be careful with geometry: a parallel ray refracts through the far focal point, and the center ray goes straight through. Do not force an object-side ray to physically pass through a near focal point before reaching the lens if that makes the diagram confusing.

Common Trap

• A positive image distance is real and inverted; a negative image distance is virtual and upright. Mixing these up flips your whole description.

Practice Problem 1: Thin-Lens Equation

Solution

Use the thin-lens equation to find the image position:

$\frac{1}{s_o}+\frac{1}{s_i}=\frac{1}{f}$

Given:

• Object distance $s_o = 15$ cm
• Focal length $f = 10$ cm (positive because it is a convex lens)

Rearranging to solve for $s_i$:

$\frac{1}{s_i}=\frac{1}{f}-\frac{1}{s_o}=\frac{1}{10}-\frac{1}{15}=\frac{3-2}{30}=\frac{1}{30}$

Therefore, $s_i = 30$ cm

The image is 30 cm from the lens on the opposite side from the object. Since $s_i$ is positive, the image is real.

To determine magnification: $M = -\frac{s_i}{s_o} = -\frac{30}{15} = -2$

The negative sign indicates the image is inverted, and the magnitude of 2 means the image is twice the size of the object.

Characteristics: The image is real, inverted, and magnified by a factor of 2.

Practice Problem 2: Ray Diagrams and Image Formation

Solution

First, use the thin-lens equation:

$\frac{1}{s_o}+\frac{1}{s_i}=\frac{1}{f}$

Given:

• Object distance $s_o = 5$ cm
• Focal length $f = 10$ cm

Rearranging to solve for $s_i$:

$\frac{1}{s_i}=\frac{1}{f}-\frac{1}{s_o}=\frac{1}{10}-\frac{1}{5}=\frac{1-2}{10}=\frac{-1}{10}$

Therefore, $s_i = -10$ cm

The negative value for $s_i$ indicates the image is virtual and forms on the same side of the lens as the object.

For magnification: $M = -\frac{s_i}{s_o} = -\frac{-10}{5} = 2$

The positive value indicates the image is upright, and the magnitude of 2 means the image is twice the size of the object.

Ray diagram analysis shows:

1. A ray parallel to the principal axis refracts through the far-side focal point
2. A ray through the center of the lens continues straight
3. A ray aimed toward the far-side focal point refracts parallel to the principal axis

These refracted rays diverge on the far side of the lens, so extend them backward to the object side. They appear to come from a point 10 cm from the lens on the same side as the object. This confirms the image is virtual, upright, and magnified.

Characteristics: The image is virtual, upright, and magnified by a factor of 2.

Common Misconceptions

• "All lens images can be projected on a screen." Only real images, where light rays actually meet, can land on a screen. Virtual images cannot.
• "A convex lens always makes an enlarged or real image." The result depends on object distance. Beyond the focal point it makes a real image; inside the focal length it makes an upright, enlarged virtual image; at $2f$ the image is the same size.
• "Concave lenses can form real images." Concave (diverging) lenses always form virtual, upright, reduced images regardless of object distance.
• "A negative focal length means I made a mistake." A negative $f$ is correct for concave lenses. Negative signs carry physical meaning, not errors.
• "Magnification greater than 1 always means a real image." Magnification size and image type are separate. A magnifier inside the focal length gives $|M|>1$ but a virtual image.
• "The ray through the focal point and the parallel ray are the same trick." They are different rays. A parallel ray bends to go through the focal point, while a ray through the focal point bends to come out parallel.

Related AP Physics 2 Guides

• 13.1 Reflection
• 13.2 Images Formed by Mirrors
• 13.3 Refraction