Concave Lens

A concave lens is a transparent optical lens that is thinner at the center and thicker at the edges.

• It causes parallel light rays to spread out (diverge) after passing through it.
• Therefore, it is also called a diverging lens.
• It forms a virtual, erect, and diminished (smaller) image for a real object.
• Its focal length is considered negative under the Cartesian sign convention.
• It is commonly used in spectacles for correcting myopia (short-sightedness).
• A concave lens looks like this: )( when viewed from the side.

Example of Concave Lens

• Eyeglasses used to correct nearsightedness (myopia).
• Galilean telescopes, where a concave lens is used as the eyepiece.
• Optical systems that use concave lenses to diverge or control light.
• Laser beam expanders, where concave lenses can help expand a laser beam.

Parts of Concave Lens

A concave lens, like other lenses, consists of several parts and features that contribute to its optical properties. Here are the main parts of a concave lens:

Focal Length of Concave lens

• The focal length of a lens is the distance between its optical center and principal focus.
• For a concave lens, the focal length is considered negative, indicating divergence of light.

Optical Center

• The optical center is a point on the principal axis of a lens through which a light ray passes without significant deviation.
• It is denoted by o.

Focus of Concave Lens

• The principal focus of a concave lens is the point on the principal axis from which parallel rays of light appear to diverge after passing through the lens.
• The virtual principal focus of a concave lens lies on the same side of the lens as the incident light.

Center of Curvature

• Each curved surface of a lens is part of a sphere. The center of that sphere is called the center of curvature.
• A lens generally has two centers of curvature, denoted by C₁ and C₂, corresponding to its two surfaces.

Principal Axis

• The principal axis is the straight line passing through the optical center and the centers of curvature of the lens surfaces.

Types of Concave lens

Concave lenses, also known as diverging lenses, come in various shapes and forms depending on their specific applications and designs. The main types of concave lenses include:

1. Biconcave Lens

• Biconcave Lens is characterized by two curved surfaces, both of which are concave.
• It is symmetrical, with the curves facing each other.
• This lens is thinner at the center than at the edges, and it diverges parallel rays of light.

2. Plano Concave Lens

• Planoconcave lens is a type of lens with one flat (plano) surface and one concave surface.
• The flat side is typically facing away from the center of the lens.
• It is a diverging lens, meaning it causes parallel rays of light to spread out.

3. Convexo-Concave Lens

• Convexo-concave lens has two different surfaces one as convex surface and other as concave surface.
• The convex side is typically facing outward.
• This lens type can be thought of as a combination of a convex lens and a concave lens, and its optical properties are determined by the radii of curvature of its surfaces.

Double Concave Lens

• "Double Concave Lens" is interchangeable with the biconcave lens.
• It refers to a concave lens with two curved surfaces that are both concave.
• This lens type is frequently used for its diverging properties and is thinner in the middle than at the edges.

Properties

Concave lenses, being diverging lenses, possess several distinctive properties. Here are the key properties of concave lenses:

• Diverging Nature: The primary property of a concave lens is its ability to diverge or spread out parallel rays of light. This is in contrast to converging lenses (convex lenses) that focus parallel rays to a point.
• Virtual Principal Focus: Concave lenses have a virtual principal focus. When parallel rays of light pass through a concave lens, they appear to diverge from a virtual point on the same side as the incident light. The focal point is considered virtual for a concave lens.
• Negative Focal Length: The focal length of a concave lens is negative. This indicates the diverging nature of the lens and is a crucial parameter in lens formula calculations.
• Virtual Images: Images formed by concave lenses are virtual. They cannot be projected onto a screen and are perceived by the observer as if the light is coming from a virtual source.
• Upright and Diminished Images: Virtual images formed by a concave lens are upright and diminished in size compared to the object. The extent of diminishment depends on the position of the object relative to the focal point.
• Focal Point and Focal Length Relationship: The focal point of a concave lens is located on the same side as the incident light. The focal length is measured from the optical center of the lens to this virtual focal point and is considered negative.

Sign Convention of Concave Lens

Following are the Sign Conventions for the Concave lens:

Focal Length(f): The Focal length for the Concave lens is Negative. This is because a concave lens diverges light, and the focal point is on the same side as the incident light.

Object Distance(u): Object distance is measured from the object to the lens. If the object is on the same side as the incident light (the side from which light approaches the lens), it is taken as negative.

Image Distance(v): Image distance is the distance between the image to the lens. For images formed on the same side as the incident light, the distance is considered negative (virtual image).

Height(h): Heights measured upwards are considered positive, and heights measured downwards are considered negative.

Ray Diagram of Concave Lens

Ray Diagram is a graphical representation used to understand the image formation by lenses. Let's draw a ray diagram for a concave lens:

Ray diagram for Object at Infinity:

• If the object is at an infinite distance from the concave lens (essentially parallel rays), the diverging rays continue to spread out after passing through the lens.
• The extended rays appear to diverge from a virtual focal point on the same side as the object.
• The image formed is virtual, upright, and diminished.

Ray Diagram for Object between Infinity and Optical Centre:

• The diverging rays, when extended backward, appear to diverge from a point on the same side as the object.
• The correct description is that the image formed is virtual, erect, and diminished (smaller than the actual object).

Table for Image formation in Concave lens

Concave Lens Formula

The lens formula is a mathematical relationship that describes the relationship between the object distance (u), the image distance (v), and the focal length (f) of a lens. The formula is applicable to both convex and concave lenses. For the concave lens , focal length is considered as negative.

The lens formula is given by:

(1/f)= (1/v)-(1/u)

Where:

• f represents focal length of the lens.
• v represents the image distance (distance from the lens to the image formed).
• u represents the object distance (distance from the lens to the object).

Concave Lens Magnification Formula

The magnification formula relates the height of an image to the height of an object in optics.

For a lens, including a concave lens, the magnification (often denoted as "m") is given by the following formula:

m = h_image/h_object

Where:

• m represents the magnification.
• h_image represents the height of the image formed by the lens.
• h_object represents the height of the object.

For a concave lens, magnification is positive because the image is virtual and erect. Since the image is diminished, the magnitude of magnification is less than 1.

The magnification formula can also be expressed in terms of object distance (u) and image distance (v) for both convex and concave lenses.

The formula is as follows:

m = v/u

Where:

• m represents the magnification.
• u represents the object distance (distance from the object to the lens).
• v represents the image distance (distance from the image to the lens).

The negative image distance v indicates that the image is on the same side as the object, not negative magnification.

Power of Concave Lens

Power of Concave Lens is the ability of concave lens to diverge the incident rays. The formula for Power of concave lens is given by

P = 1/(f)

where,

• P is Power
• F is focal length of the lens in m

For a concave lens, f is already negative, so its power is automatically negative.

Applications of Concave Lens

Concave lenses, also known as diverging lenses, have various applications in optics and technology. Their ability to diverge parallel rays of light is utilized in different contexts. Here are some common applications of concave lenses:

Corrective Eyewear: Concave lenses are used in eyeglasses to correct vision problems such as nearsightedness (myopia). By causing parallel rays of light to diverge before entering the eye, a concave lens helps bring the image into focus on the retina.

Projectors: Concave lenses are used in projectors to spread out light rays. This is especially useful for creating a wider and larger image projection on a screen.

Camera Systems: Certain optical systems, including cameras, use concave lenses to diverge light rays. In some camera designs, concave lenses are used to correct aberrations and distortions.

Flashlights and Car Headlights: Some flashlights and car headlights use concave lenses to spread out light, providing a wider and more even illumination.

Binoculars and Spotting Scopes: In optical instruments like binoculars and spotting scopes, concave lenses may be used in combination with convex lenses to enhance the overall performance of the system.

Galilean Telescopes: Galilean telescopes use a convex objective lens and a concave eyepiece. The concave eyepiece helps increase the eye relief and provides a wider field of view.

Solved Examples

Example 1: An object is placed 20 cm in front of a concave lens with a focal length of -15 cm. Determine the image distance and describe the characteristics of the image formed.

Solution:

Given

• Object distance (u) = -20 cm (negative since the object is in front of the lens)
• Focal length (f) = -15 cm (negative for a concave lens)

Using the lens formula:

(1/f)= (1/v)-(1/u)

Substitute the known values:

(1/-15) = (1/v) - (1/-20)

Solve for v:

1/v = (1/-15) - (1/20)

1/v = -(7/60)

v = -8.57 cm

The image distance (v) is -8.57 cm. The negative sign indicates that the image is formed on the same side as the incident light. The image which we get will be virtual, upright, and diminished.

Example 2: A concave lens has a focal length (f) of -15 cm. An object is placed 20 cm in front of the lens. Determine the image distance (v).

Solution:

Given:

u = -20 cm

f = -15 cm

Using the lens formula:

(1/f) = (1/v)-(1/u)

Substitute the known values:

(1/-15) = (1/v)-(1/-20)

Solve for v:

1/v = (-1/15)-(1/20)

1/v = -(7/60)

v = -8.57 cm

So, the image distance (v) is -10 cm.

Example 3: A concave lens with a focal length (f) of -12 cm is used to form an image. An object is placed 24 cm in front of the lens. Determine the image distance (v). Calculate the height of the image formed if the object has a height of 6 cm.

1. Determine the image distance (v).
2. Calculate the height of the image formed if the object has a height of 6 cm.

Solution:

1. Calculation of Image Distance (v):

u = -24 cm

f = -12 cm

Using the lens formula:

(1/f)= (1/v)-(1/u)

Substitute the known values:

(1/-12)= (1/v) -(1/-24)

Solve for v:

1/v = (1/-12)-(1/24)

1/v = -(3/24)

v = -8 cm

So, the image distance (v) is -8 cm.

2. Calculation of Image Height:

Given:

h_object = 6 cm

The magnification (m) is given by the formula:

m = v/u

Substitute the values:

m = (−8/−12)

The image height (h_image) is related to the object height by the magnification:

h_image = m × h_object

Substitute the values:

h_image = (8/12) × 6 = 4 cm

So, the height of the image is 4 cm.

Example 4: Suppose you have a concave lens with an object placed 20 cm in front of it, and the image is formed at 10 cm. Find the focal length.

Solution:

Given:

u = -20 cm

v = -10 cm

Using the lens formula:

(1/f) = (1/v)-(1/u)

Substitute the known values:

(1/f) = (1/-10)-(1/-20)

(1/f) = -1/20

So, the focal length is -20 cm.

Practice Questions

Q1. A concave lens has an object placed 15 cm in front of it and forms a virtual image 10 cm from the lens. Find the focal length.

Q2. For a concave lens, an object is placed 25 cm in front of it, and the resulting image is formed at −10 cm. Determine the focal length.

Q3. A concave lens is used to form an image. If the object distance is -18 cm and the image distance is -9 cm, calculate the focal length.

Q4. Suppose a concave lens forms a virtual image when the object is placed 12 cm in front of it, and the image distance is -24 cm. Determine the focal length.

Q5. For a concave lens, the object distance is -30 cm, and the image is formed at -15 cm. Find the focal length.