Thin lenses Revision Notes

A lens is a transparent optical device made of glass or plastic that refracts (bends) light rays passing through it. Thin lenses have a thickness small compared to their focal length, allowing simplification in analysis.

When light passes from one medium to another (e.g., air to glass), it changes speed, causing the light ray to bend at the interface. This bending is called refraction and is governed by Snell's law.

Lenses use refraction at their curved surfaces to change the direction of light rays, affecting how images are formed. The shape of the lens determines whether it converges or diverges light.

• Refraction changes light direction due to speed change in different media.
• Thin lenses simplify ray tracing by assuming negligible thickness.
• Lens shape determines converging or diverging behaviour.

A converging lens is thicker in the middle than at the edges. When a parallel beam of light rays (such as from a distant object) enters a converging lens, the rays are refracted towards the principal axis and meet at a point called the principal focus (focal point) on the opposite side of the lens.

This focal point is real because the refracted rays physically converge there and can be projected onto a screen.

The distance from the centre of the lens to the principal focus is the focal length (f), measured in metres (m).

• Converging lenses focus parallel rays to a real focal point on the opposite side.
• The focal length is the distance from lens centre to focal point.
• Real focal points allow image projection onto screens.

A diverging lens is thinner in the middle than at the edges. When a parallel beam of light rays passes through a diverging lens, the rays spread out (diverge) after refraction.

These diverging rays appear to come from a point on the same side of the lens as the incoming light. This point is called the virtual principal focus (virtual focal point).

Because the rays do not actually meet, the focal point is virtual and cannot be projected onto a screen. It is found by tracing the diverging rays backward until they intersect.

• Diverging lenses cause parallel rays to spread out after refraction.
• Virtual focal point lies on the same side as incoming light.
• Virtual focal points cannot form real images on screens.

The bending of light rays by lenses occurs due to refraction at the curved surfaces. The curvature causes rays entering at different distances from the principal axis to refract by different amounts.

In converging lenses, the shape causes rays to bend towards the axis, focusing them to a point.

In diverging lenses, the shape causes rays to bend away from the axis, making them appear to diverge from a point on the incoming side.

• Lens curvature determines the direction of ray bending.
• Refraction at curved surfaces focuses or spreads rays.
• Physical cause is the change in light speed and direction at interfaces.

To illustrate how lenses affect parallel rays, draw the principal axis as a horizontal line.

For a converging lens, draw several parallel rays approaching the lens from the left. After the lens, draw the rays refracted so that they meet at the focal point on the right side.

For a diverging lens, draw parallel rays approaching from the left. After the lens, draw the rays diverging. Extend these rays backward with dashed lines to meet at the virtual focal point on the left side of the lens.

• Draw principal axis as baseline.
• For converging lens, refracted rays meet at real focal point on opposite side.
• For diverging lens, refracted rays diverge; backward extensions meet at virtual focal point on same side.

The principal axis is an imaginary straight line that passes through the optical centre of a lens and is perpendicular to its surfaces. For a thin lens, this line runs through the centre of the lens and extends on both sides. It serves as a reference line for describing the path of light rays and the positions of important points such as the focal points and the lens centre.

The principal axis directly affects the result because it helps define the geometry of the lens system and the behaviour of light rays that are parallel or inclined to this axis.

• Imaginary straight line through the lens centre.
• Perpendicular to the lens surfaces.
• Reference line for ray diagrams and focal points.

The principal focus, also called the focal point, is the point on the principal axis where light rays that are initially parallel to the principal axis either converge (for converging lenses) or appear to diverge from (for diverging lenses) after passing through the lens.

For a converging (convex) lens, parallel rays of light refract through the lens and meet at the principal focus on the opposite side of the lens.

For a diverging (concave) lens, parallel rays appear to spread out after refraction, and if these rays are extended backwards, they appear to come from the principal focus on the same side as the incoming light.

• Point where parallel rays meet or appear to meet.
• Located on the principal axis.

The focal length (denoted by f) is the distance between the optical centre of the lens and its principal focus along the principal axis. It is a positive value for converging lenses and negative for diverging lenses.

The focal length determines the lens's ability to converge or diverge light. A shorter focal length means stronger convergence or divergence.

The unit of focal length is metres (m), but centimetres (cm) or millimetres (mm) are often used in practical contexts.

• Distance from lens centre to principal focus.
• Positive for converging lenses, negative for diverging lenses.

To construct a ray diagram for a thin lens, use the principal axis, principal focus, and optical centre as reference points. Three principal rays are typically drawn:

A ray parallel to the principal axis refracts through the lens and passes through (or appears to come from) the principal focus.

A ray passing through the optical centre continues in a straight line without deviation (assuming a thin lens). A ray passing through the principal focus emerges parallel to the principal axis after refraction through the lens. These rules allow determination of the image position, size, orientation, and type (real or virtual).

• Use principal axis as baseline.
• Draw rays using principal focus and optical centre.

If the focal length of a converging lens is 20 cm, and parallel rays of light enter the lens, they will converge at a point 20 cm from the lens along the principal axis.

Formula: Focal length, f = distance from lens to principal focus

Example: f = 20 cm = 0.20 m This means the principal focus is 0.20 m from the lens centre on the side opposite to the incoming light rays (for a converging lens). Common error: Confusing focal length with radius of curvature or mixing units. Always ensure units are consistent and focal length is measured along the principal axis from the optical centre.

• f = distance from lens centre to principal focus.
• Example: f = 20 cm means focus is 20 cm from lens.

A converging lens, also called a convex lens, is thicker at the centre than at the edges. It causes parallel rays of light incident on it to refract and converge to a single point called the principal focus or focal point (F).

The distance from the centre of the lens (optical centre) to the focal point is the focal length (f), measured in metres (m).

There are two focal points, one on each side of the lens, equidistant from the optical centre along the principal axis (a horizontal line through the centre of the lens).

• Converging lens: thicker at centre, converges light rays.
• Principal axis: horizontal line through lens centre.
• Focal points (F): points where parallel rays converge.

To construct a ray diagram for an object placed in front of a converging lens, three principal rays are drawn from the top of the object (usually represented as an arrow):

• Ray 1: Drawn parallel to the principal axis; after refraction through the lens, it passes through the focal point on the opposite side.

1. Draw the converging lens as a vertical line and mark the principal axis horizontally through its centre.

2. Mark the focal points (F) on both sides at a distance equal to the focal length.

3. Place the object (an arrow) upright on the principal axis to the left of the lens (object side). The base of the arrow lies on the principal axis, and the arrow points upwards representing the object height and orientation (upright). 4. From the top of the object, draw the three principal rays as described above, using a ruler and pencil for accuracy. 5. The point where the refracted rays intersect on the opposite side of the lens is the top of the image. Draw the image arrow from the principal axis to this point.

• Lens drawn as vertical line; principal axis horizontal.
• Focal points marked on both sides at focal length distance.
• Object drawn as upright arrow on object side.

The image formed by a converging lens when the object is placed beyond the focal length has the following properties:

It is real, meaning the light rays actually converge at the image location and the image can be projected onto a screen.

It is inverted (upside down) relative to the object orientation, as shown by the arrow pointing downwards in the diagram compared to the object arrow pointing upwards. - The size of the image can be larger, smaller, or the same size as the object depending on the object's distance from the lens: closer to the lens (but beyond the focal length) produces a larger image; farther away produces a smaller image. - The image is located on the opposite side of the lens from the object.

• Real image: formed by actual convergence of rays; can be projected.
• Image is inverted relative to the object.

The converging lens bends (refracts) light rays towards the principal axis due to the change in speed of light in the lens material compared to air. This refraction causes parallel rays to meet at the focal point.

When an object is placed beyond the focal length, the refracted rays converge to form a real image. This principle is used in devices such as cameras, projectors, and the human eye to form clear images on a screen or retina.

Understanding ray diagrams helps predict image position and size, which is essential for designing optical instruments.

• Refraction bends light rays towards principal axis.
• Real images form when rays converge after lens.

When light passes through a thin lens, it forms an image of an object. This image can be described using specific terms that relate to its size, orientation, and nature. Understanding these terms helps in analysing how lenses form images and predicting their properties based on object position relative to the lens.

• Image: The reproduction of an object formed by light rays after passing through a lens.
• Object: The original item whose image is formed by the lens.

The size of the image compared to the object is a key characteristic. It is described as:

Enlarged: The image is larger than the object. This occurs when the image height is greater than the object height.

Same size: The image height equals the object height; the image and object are of equal size, often occurring at specific object distances such as at twice the focal length for a converging lens (2f). Diminished: The image is smaller than the object, with image height less than object height. This typically happens when the object is far from the lens (beyond 2f).

• Image size is measured as the height of the image compared to the object height.

Orientation describes whether the image is the same way up as the object or flipped upside down:

Upright: The image has the same orientation as the object; if the object is standing upright, so is the image.

Inverted: The image is upside down relative to the object; the top of the object corresponds to the bottom of the image. Orientation is important in applications like photography and vision correction.

• Orientation is determined by the direction of the image relative to the principal axis.

Images formed by lenses can be classified as real or virtual based on how the light rays behave:

Real image: Formed where actual light rays converge after passing through the lens. Real images can be projected onto a screen because light physically passes through the image location.

Virtual image: Formed where light rays appear to diverge from a point behind the lens. The rays do not actually meet but appear to do so when extended backward. Virtual images cannot be projected onto a screen as no real light passes through the image location.

• Real images are always inverted for single lenses.
• Virtual images are always upright.
• Converging lenses can form both real and virtual images depending on object position.

The characteristics of an image depend on the object’s position relative to the lens’s focal points:

When the object is beyond twice the focal length (2f) of a converging lens, the image is real, inverted, diminished, and formed between f and 2f.

At 2f, the image is real, inverted, and the same size as the object, formed at 2f on the other side of the lens. Between f and 2f, the image is real, inverted, and enlarged, formed beyond 2f on the other side. At the focal point (f), no image is formed as rays emerge parallel. Between the lens and the focal point, the image is virtual, upright, and enlarged, appearing on the same side as the object.

• Image size and orientation result from the geometry of ray convergence or divergence.
• Real images can be captured on a screen; virtual images cannot.

A virtual image is an image formed when the outgoing rays from a point on an object diverge after passing through a lens or reflecting from a mirror. When these diverging rays are extended backward (traced behind the lens or mirror), they appear to meet at a point. This point is where the virtual image is located.

Unlike a real image, the light rays do not actually converge at the virtual image position; instead, the image is formed by the apparent intersection of the backward extensions of the rays.

Because the rays do not physically meet, a virtual image cannot be captured on a screen placed at the image location. Instead, it can only be seen by looking into the lens or mirror.

• Virtual image: formed by backward extension of diverging rays.

A diverging lens causes parallel rays of light to spread out (diverge) after passing through it. When these diverging rays are traced backward in a straight line, they appear to originate from a point on the same side of the lens as the object.

This point is the virtual image formed by the diverging lens. The image is upright, diminished (smaller than the object), and located between the lens and the object.

Because the rays do not actually meet, the virtual image cannot be projected onto a screen. It is only visible by looking through the lens.

• Diverging lens causes rays to spread out.
• Backward extensions of rays meet at virtual image point.

To draw a ray diagram showing a virtual image formed by a diverging lens:

1. Draw the principal axis as a horizontal line through the centre of the lens.

2. Mark the optical centre of the lens and the focal points (F) on both sides; for a diverging lens, focal length (f) is negative, and focal points are virtual points on the object side of the lens (left side if object is on left). 3. From the top of the object, draw a ray parallel to the principal axis towards the lens. After refraction through the diverging lens, this ray diverges away from the principal axis. Extend this refracted ray backward as a dotted line passing through the focal point on the object side. 4. Draw a second ray from the top of the object passing through the optical centre of the lens; this ray continues straight without deviation. 5. The point where the backward extensions of the refracted rays meet is the virtual image location. 6. Draw the image as an upright arrow at this point, smaller than the object.

• Principal axis and optical centre drawn horizontally.
• Focal points marked on object side for diverging lens.
• Ray parallel to axis refracted diverging; backward extension passes through focal point.

A real image is formed where actual light rays converge and can be displayed on a screen placed at that location.

In contrast, a virtual image is formed where rays only appear to come from but do not physically meet. Since no real light converges at the virtual image point, placing a screen there will not capture the image.

The eye perceives the virtual image because the brain interprets the diverging rays as if they originated from the virtual image point behind the lens or mirror.

• Real image: rays physically converge; can be projected.

| Property | Real Image | Virtual Image |

|------------------------|--------------------------------|---------------------------------|

| Formation | Rays actually converge | Rays diverge; backward extensions converge | | Projection | Can be projected onto a screen | Cannot be projected onto a screen | | Orientation | Inverted | Upright | | Size | Can be smaller, same, or larger| Usually smaller (for diverging lens) | | Location | On opposite side of lens from object | On same side as object |

A converging lens, also called a convex lens, is thicker at the centre than at the edges. It refracts (bends) light rays towards the principal axis, causing parallel rays to converge at the principal focus (focal point) on the opposite side of the lens.

The focal length (f) is the distance from the lens centre to the principal focus. It is positive for converging lenses and measured in metres (m).

When an object is placed beyond the focal length, the lens forms a real image on the opposite side. However, when the object is placed closer to the lens than the focal length, the lens forms a virtual image on the same side as the object.

• Converging lens = convex shape, thicker centre.
• Focal length (f) is positive and fixed for a given lens.

A virtual image is formed by a converging lens only when the object lies between the lens and its focal point (object distance u < focal length f).

In this case, the refracted rays diverge on the opposite side of the lens, so they do not actually meet. However, if these rays are extended backwards (behind the lens on the object side), they appear to meet at a point where the virtual image is formed.

The virtual image is upright (same orientation as the object), magnified (larger than the object), and cannot be projected onto a screen because the rays do not actually converge there.

• Object inside focal length (u < f) → virtual image.
• Virtual image is on the same side as the object.

To draw a ray diagram showing a virtual image formed by a converging lens, follow these steps:

1. Draw the principal axis as a horizontal straight line.

2. Represent the lens as a vertical line at the centre, marking the focal points (F) on both sides at distance f from the lens centre (O). Label the focal points as F (right side) and F' (left side). The object is placed between O and F' on the left side (object side). Represent the object as an arrow standing on the principal axis, with the arrowhead above the axis.

• Draw the principal axis and lens centre (O).

Draw two rays from the top of the object: 1. The first ray travels parallel to the principal axis towards the lens. After refraction through the lens, this ray diverges as if coming from the focal point F' on the object side. Extend this refracted ray backwards (dashed line) behind the lens to pass through F'.

2. The second ray passes straight through the optical centre O of the lens without deviation.

The point where the two backward-extended rays meet behind the lens locates the top of the virtual image. Draw the image arrow from the principal axis to this point.

• Ray 1: Parallel to axis → refracted ray appears to come from F' (extend backwards).

From the ray diagram, the virtual image formed by a converging lens when the object is inside the focal length has the following properties:

Location: On the same side as the object, behind the lens.

Orientation: Upright (same way up as the object). This contrasts with real images, which are inverted. Size: Larger than the object (magnified). Nature: Virtual, meaning the rays do not actually converge but only appear to do so when extended backwards. This image cannot be projected onto a screen.

A magnifying glass is a simple optical instrument that uses a single convex lens to enlarge the appearance of small objects. It helps the eye see details that would otherwise be difficult to observe with the naked eye.

The lens used is a converging lens, meaning it bends parallel rays of light towards a focal point. This property is essential for producing a magnified image.

• Magnifying glass uses a convex (converging) lens.
• Purpose: to enlarge the apparent size of an object for clearer viewing.

When light rays from an object pass through a convex lens, they are refracted (bent) towards the principal axis. The amount of bending depends on the curvature of the lens surfaces and the refractive index of the lens material.

For a magnifying glass, the object is placed closer to the lens than its focal length (distance from lens to principal focus). Under this condition, the lens forms a virtual, upright, and enlarged image on the same side of the lens as the object.

This virtual image cannot be projected onto a screen because the refracted rays diverge, but the eye perceives them as coming from a larger object located behind the lens.

• Light rays bend towards the principal axis when passing through a convex lens.
• Object placed within the focal length produces a virtual, upright, enlarged image.
• Virtual image appears on the same side as the object and cannot be caught on a screen.

To draw a ray diagram for a magnifying glass: 1. Draw the principal axis as a horizontal line. 2. Mark the lens as a vertical line and indicate the focal points (F) on both sides at the focal length distance from the lens.

3. Place the object between the lens and the focal point on the left side. 4. Draw one ray from the top of the object parallel to the principal axis; after refraction through the lens, this ray passes through the focal point on the right side.

5. Draw a second ray from the top of the object passing through the centre of the lens; this ray continues straight without bending. 6. The two refracted rays diverge on the right side, so extend them backward with dotted lines to meet behind the object on the left side. The intersection point is the top of the virtual image, which is upright and larger than the object.

• Principal axis and focal points must be marked.
• Object placed between lens and focal point.
• Refracted rays diverge; backward extensions meet to form virtual image.
• Image is upright and magnified.

Magnification (M) is the ratio of the image height (h_i) to the object height (h_o): M = h_i / h_o. For a magnifying glass, magnification is greater than 1, meaning the image appears larger than the object.

The maximum magnification occurs when the object is as close as possible to the lens without causing eye strain, typically just inside the focal length.

The eye focuses on the virtual image formed at a comfortable viewing distance, allowing detailed observation without the need for accommodation to focus on very close objects.

• Magnification M = image height / object height.
• Object placed within focal length yields M > 1.
• Comfortable viewing distance achieved by virtual image formation.

Magnifying glasses are commonly used to examine small objects such as insects, text, or fine details in materials.

Limitations include the size of the lens and the maximum magnification achievable, which depends on the focal length and quality of the lens.

Using a magnifying glass requires holding the object within the focal length of the lens and adjusting the distance to achieve a clear, enlarged virtual image.

• Used for detailed observation of small objects.
• Magnification limited by lens focal length and quality.
• Proper positioning of object relative to lens is essential.

The human eye normally focuses light rays from objects onto the retina, producing a clear image. Vision defects occur when the eye's lens cannot focus light correctly on the retina.

Two common defects are: long-sightedness (hyperopia), where distant objects are seen clearly but near objects appear blurred, and short-sightedness (myopia), where near objects are clear but distant objects are blurred.

These defects result from the eye's lens focusing images either behind or in front of the retina, respectively.

• Long-sightedness: image forms behind the retina.
• Short-sightedness: image forms in front of the retina.

In long-sightedness, the eye's lens is too weak or the eyeball is too short, so light from near objects is focused behind the retina, causing blurred near vision.

A converging lens (convex lens) is used to correct this defect. It converges (brings together) the incoming light rays before they enter the eye, effectively increasing the eye's focusing power.

This adjustment moves the focal point forward so that the image forms exactly on the retina, allowing clear vision of nearby objects.

• Converging lenses are thicker at the centre than at the edges.
• They have a positive focal length and cause parallel rays to converge at the principal focus.
• Used to correct long-sightedness by converging light rays before they enter the eye.

In short-sightedness, the eye's lens is too strong or the eyeball is too long, so light from distant objects is focused in front of the retina, causing blurred distance vision.

A diverging lens (concave lens) is used to correct this defect. It diverges (spreads out) the incoming light rays before they enter the eye, effectively reducing the eye's focusing power.

This adjustment moves the focal point backward so that the image forms on the retina, allowing clear vision of distant objects.

• Diverging lenses are thinner at the centre than at the edges.
• They have a negative focal length and cause parallel rays to diverge as if from the principal focus.
• Used to correct short-sightedness by diverging light rays before they enter the eye.

Ray diagrams help visualise how lenses correct vision defects. To draw a ray diagram for vision correction:

1. Draw the eye's lens and retina as a vertical line representing the retina.

2. For long-sightedness, show light rays from a near object converging behind the retina without correction; then add a converging lens in front of the eye to bring rays to focus on the retina after passing through the eye's lens. 3. For short-sightedness, show light rays from a distant object converging in front of the retina without correction; then add a diverging lens in front of the eye to spread rays so they focus on the retina after the eye's lens.

• Use principal rays: one parallel to the principal axis refracted through the focal point, one through the centre of the lens undeviated.
• The corrected image forms on the retina, producing clear vision.

Converging (convex) lenses have positive focal lengths and bring parallel rays together; they correct long-sightedness by moving the focal point forward onto the retina.

Diverging (concave) lenses have negative focal lengths and spread parallel rays apart; they correct short-sightedness by moving the focal point backward onto the retina.

Both types of lenses adjust the path of incoming light to compensate for the eye's focusing error, enabling clear images on the retina.

• Converging lens: thicker centre, corrects hyperopia.
• Diverging lens: thinner centre, corrects myopia.
• Lens power is measured in dioptres (D), where power P = 1/f (f in metres).