Understanding Lens Power and Focal Length

The question asks us to determine the type of lens and its focal length given its power. The power of a lens is a measure of how much it converges or diverges light. It is related to the focal length of the lens.

Defining Lens Power

Lens power (\(P\)) is defined as the reciprocal of the focal length (\(f\)). The standard unit for lens power is the Dioptre (D). For the power to be in Dioptres, the focal length must be measured in metres.

The formula relating power and focal length is:

\(P = \frac{1}{f}\)

where \(P\) is in Dioptres and \(f\) is in metres.

Sign Convention for Lenses

The sign of the power and focal length tells us about the type of lens:

• A positive power (\(P > 0\)) indicates a converging lens. For lenses commonly used, a converging lens is a convex lens.
• A negative power (\(P < 0\)) indicates a diverging lens. For lenses commonly used, a diverging lens is a concave lens.
• A positive focal length (\(f > 0\)) corresponds to a converging (convex) lens.
• A negative focal length (\(f < 0\)) corresponds to a diverging (concave) lens.

Calculating Focal Length

We are given that the power of the lens is \(P = +2.0\) Dioptre. We can use the formula \(P = \frac{1}{f}\) to find the focal length \(f\).

Rearranging the formula to solve for \(f\):

\(f = \frac{1}{P}\)

Substituting the given power:

\(f = \frac{1}{+2.0 \text{ D}}\)

\(f = +0.5 \text{ metres}\)

Determining Lens Type and Focal Length

From the calculation, we found the focal length \(f = +0.5\) metres. Since the focal length is positive, the lens is a converging lens.

As per the sign convention, a lens with positive power and positive focal length is a convex lens.

Therefore, the lens is convex and has a focal length of 0.5 metre.

Analyzing the Statements

Let's examine each given statement based on our findings:

1. The lens is concave and has a focal length of 0.5 metre.
   This statement is incorrect. The lens is convex, not concave, because the power is positive.
2. The lens is convex and has a focal length of 2.0 metre.
   This statement is incorrect. While the lens is convex, its focal length is 0.5 metre, not 2.0 metre.
3. The lens is convex and has a focal length of 0.5 metre.
   This statement is correct. The lens is convex because the power is positive, and the calculated focal length is 0.5 metre.
4. The lens is concave and has a focal length of 2.0 metre.
   This statement is incorrect. The lens is convex, not concave, and the focal length is 0.5 metre, not 2.0 metre.

Based on our analysis and calculation, the statement that correctly describes the lens is that it is convex and has a focal length of 0.5 metre.

Revision Table: Lens Properties Summary

Additional Information: Applications of Lenses

Understanding lens power and focal length is crucial in many applications:

• Eyeglasses: Lenses are used to correct vision problems like myopia (nearsightedness), hyperopia (farsightedness), astigmatism, and presbyopia. The power of the lens required depends on the specific refractive error.
• Cameras: Lenses focus light onto the sensor or film to create an image.
• Microscopes: Combine multiple lenses to produce highly magnified images of small objects.
• Telescopes: Use lenses (refracting telescopes) or mirrors (reflecting telescopes) to view distant objects like stars and planets.
• Projectors: Lenses project images onto a screen.

The unit Dioptre is very convenient for ophthalmologists and optometrists as lens powers can be added directly when lenses are placed in contact.