Lens Sign Convention

A lens in optics is a transparent device with curved surfaces that refract light. It can be converging or diverging based on its shape. The lens sign convention, or Cartesian sign convention, helps determine the nature, size, and position of images formed by lenses accurately. It clarifies the positive and negative signs for object and image distances, focal lengths, and magnifications, crucial for optical calculations and predictions.

What is the Lens Sign Convention?

The lens sign convention is a set of rules used in geometrical optics to describe the behavior of light rays as they pass through lenses. These conventions are used to determine the sign conventions for various quantities such as object distance, image distance, focal length, and magnification. The conventions are typically employed in ray diagrams and equations to predict the characteristics of images formed by lenses.

What is a Lens in Optics?

In optics, a lens is a transparent optical device with curved surfaces that refract light, causing it to converge or diverge. There are two main types of lenses based on their shape:

• Convex Lens (Converging Lens) : This lens is thicker in the center than at the edges. It converges parallel rays of light to a focal point after passing through the lens. Convex lenses are commonly used in magnifying glasses, camera lenses, and eyeglasses to correct hypermetropia (farsightedness).
• Concave Lens (Diverging Lens): This lens is thinner in the center than at the edges. It diverges parallel rays of light, making them appear to diverge from a virtual focal point behind the lens. Concave lenses are often used in correcting myopia (nearsightedness).

Basics of Lens Sign Convention

The following are some terms that must be understood before starting the lens equation :

• Object Distance (u): Object distances are measured from the object to the lens along the direction of the incident light.
• Image Distance (v): Image distances are measured from the lens to the image along the direction in which the light travels after passing through the lens.
• Focal Length (f): Focal length refers to the distance between the optical center of a lens and the point where light rays converge to form a sharp image of an object when the lens is focused at infinity.
• Magnification (m): The ratio of the height of the image to the height of the object, is called magnification.

Cartesian Sign Convention

Table for the cartesian sign convention is given as follows:

Quantity	Condition	Sign
Focal Length(f)	Concave lens	Positive(+)
	Convex lens	Negative(-)
Object Distance (u)	Always	Negative(-)
Image Distance (v)	Real Image	Positive(+)
	Virtual Image	Negative(-)
Magnification (m)	Upright Image	Positive(+)
	Inverted Image	Negative(-)

Sign Conventions in Different Lenses

The signs of the distances need to be carefully considered based on the type of lens. A short comparison for sign convention between convex and concave lens has been given below:

Convex Lens Sign Convention

For a convex lens, the sign convention is usually as follows:

• If the focal point (F) is on the left, it is positive.
• If the object is on the left side, the object distance (u) is positive.
• If the image is generated on the right side, the image distance (v) is positive.

Concave Lens Sign Convention

For a concave lens, the sign convention is usually as follows:

• There is a negative focal length.
• There is a negative image distance.
• There is never a positive object distance.
• From the optical centre of the lens to the right side, all measured distances are positive.

Read more about Difference Between Concave and Convex Lens.

Lens Maker’s formula

The Lens Maker’s Formula is an equation used to relate the focal length of a lens to its physical characteristics, such as its curvature and refractive index. This formula applies specifically to thin lenses, which are lenses that have negligible thickness compared to their radii of curvature. The equation is given by:

1/f = (μ-1)(1/R₁-1/R₂)

Where,

• f is the focal length of the lens.
• μ is the refractive index of the lens material.
• R₁ is the radius of curvature of one surface of the lens.
• R₂ is the radius of curvature of the other surface of the lens.

Note: If a surface is convex (curved outward), its radius of curvature is taken as positive and if surface is concave (curved inward), its radius of curvature is taken as negative.

Read more about Lens Maker’s Formula.

Magnification Formula

The magnification formula for a lens is given by:

m = h/h′​ = −u/v​

Where,

• m represents the magnification.
• h′ is the height of the image.
• h is the height of the object.
• v is the image distance.
• u is the object distance.

Power of Lens

The power of a lens is a measure of its ability to bend light, and it’s expressed in diopters (D). The power of a lens depends on its focal length f, and the formula to calculate it is:

P = 1/f​

Where:

• P represents the power of the lens in diopters (D).
• f is the focal length of the lens in meters (m).

Conclusion: Lens Sign Convention

In conclusion, the sign convention for lenses plays a crucial role in understanding image formation. By adhering to these rules, we determine whether distances are positive or negative. Convex lenses have a positive focal length, while concave lenses have a negative focal length. These conventions guide us in analyzing optical systems and their properties.

Solved Problems on Lens Sign Convention

Example 1: An object is placed 20 cm from a converging lens with a focal length of 10 cm. Calculate the image distance and determine the nature (real or virtual), orientation (upright or inverted), and size of the image formed.

Solution:

Given:

• Object distance, u = 20 cm (positive, as it’s in front of the lens)
• Focal length, f = 10 cm (positive, as it’s for a convex lens)

Using the lens equation:

1/v + 1/u = 1/f

Substituting the given values:

1/v + 1/20 = 1/10

Now, let’s solve for v :

1/v = 1/10 – 1/20 = 1/20

So, the image distance is v = 20 cm. Now, let’s analyze the characteristics of the image:

Since the image distance (20 cm) is positive, the image is formed on the opposite side of the lens from the object, indicating it is a real image.

To determine if the image is upright or inverted, we need to find the magnification.

The magnification m is given by the formula: m=-v/u

Substituting the given values: m=-1

Since the magnification is negative, the image is inverted relative to the object and the size of the image is the same as the size of the object.

Therefore, the image formed by the lens is a real image, inverted, and the same size as the object.

Example 2: An object is placed 15 cm from a converging lens. The image formed is real, inverted, and four times the size of the object. Calculate the focal length of the lens.

Solution:

Given:

• Object distance, u = 15 cm (positive, as it’s in front of the lens)
• Magnification, m = -4 (negative because inverted image)

The magnification m is given by the formula: m=-v/u

Now, let’s solve for v :

-4 = -v/15

So, the image distance is v = 60 cm.

Now that we have both v and u, using the lens equation:

1/v + 1/u = 1/f

Substituting the given values:

1/f = 1/60 + 1/15 = 3/60 = 1/20

So, the focal length is f = 20 cm.

Example 3: A 4.00-cm tall light bulb is placed a distance of 35.5 cm from a diverging lens having a focal length of -12.2 cm. Determine the image distance and the image size.

Solution:

Given:

• Object distance, u = 35.5 cm
• Focal length, f = -12.2 cm (negative, as it’s for a concave lens)

Using the lens equation:

1/v + 1/u = 1/f

Substituting the given values:

1/v + 1/35.5 = 1/-12.2

Now, let’s solve for v:

1/v = 1/-12.2 – 1/35.5

⇒ 1/v = – [(35.5+12.2)/433.1]

⇒ 1/v = -(47.7/433.1)

⇒ 1/v =- 0.110

⇒ v = – (1/0.110)

⇒ v = -9.09 cm

So, the image distance is v=-9.09 cm.

The magnification m is given by the formula: m=-v/u

Substituting the given values: m=-(-9.09)/35.5=0.256

Since the magnification is positive, the image is upright relative to the object.

The height of image , h_i=m x h_o=0.256 x 4=1.02 cm

Therefore, the image formed by the lens is a virtual image, erect, and the size is 1.02 cm.

Lens Sign Convention: Practice Problems

Problem 1: An object is placed 15 cm in front of a converging lens with a focal length of 10 cm. Calculate the image distance and determine the nature (real or virtual), orientation (upright or inverted), and size of the image formed.

Problem 2: A 3.0 cm tall object is placed 20 cm in front of a converging lens with a focal length of 15 cm. Calculate the image distance and determine the nature (real or virtual), orientation (upright or inverted), and size of the image formed.

Problem 3: An object is placed 30 cm in front of a diverging lens with a focal length of -20 cm. Calculate the image distance and determine the nature (real or virtual), orientation (upright or inverted), and size of the image formed.

FAQs on Lens Sign Convention

What is Lens?

A lens is a piece of transparent material, typically shaped like a circle, that uses refraction to focus light rays.

What is Lens Sign Convention Table?

Table for lens sign convention is given below:

Parameter	Sign Convention
Object Distance (u)	Positive in front of the lens (real object)
	Negative behind the lens (virtual object)
Image Distance (v)	Positive on the opposite side of the incident light (real image)
	Negative on the same side as the incident light (virtual image)
Focal Length (f)	Positive for converging lens (convex lens)
	Negative for diverging lens (concave lens)

How to Apply Lens Sign Convention in Calculations?

Here’s how you can apply the lens sign convention in calculations,

• Identify the type of lens
• Determine the sign of focal length (f)
• Assign signs to Object Distance (u) and Image Distance (v).

Is v positive or negative in a convex lens?

Image distances (v) in a convex mirror are always positive since the image is always created behind the mirror.

Is v negative in concave lens?

For a concave lens, the image distance (V) is always negative. This is due to the fact that a concave lens’s focal length (f) is always negative.