Calculating Net Power of Lens Combination

This question asks us to find the net power of a combination of two convex lenses placed in contact. To solve this, we first need to understand the concept of the power of a lens and how it relates to focal length, and then apply the rule for combining powers when lenses are in contact.

Understanding Lens Power and Focal Length

The power of a lens is a measure of its ability to converge or diverge light rays. It is defined as the reciprocal of the focal length \(f\). The standard unit for lens power is the dioptre (D), which is defined as the power of a lens with a focal length of 1 meter.

The formula for lens power \(P\) is:

Where \(f\) is the focal length in meters.

For convex lenses, the focal length is considered positive, resulting in positive power. For concave lenses, the focal length is negative, resulting in negative power.

Calculating Individual Lens Powers

We are given the focal lengths of two convex lenses:

• Lens 1: \(f_1 = 50\) cm
• Lens 2: \(f_2 = 25\) cm

To calculate the power in dioptres, we must convert the focal lengths from centimeters to meters:

• \(f_1 = 50 \text{ cm} = 0.50 \text{ m}\)
• \(f_2 = 25 \text{ cm} = 0.25 \text{ m}\)

Now, we can calculate the power of each lens using the formula \(P = 1/f\):

• Power of Lens 1, \(P_1 = \frac{1}{f_1} = \frac{1}{0.50 \text{ m}} = +2 \text{ D}\)
• Power of Lens 2, \(P_2 = \frac{1}{f_2} = \frac{1}{0.25 \text{ m}} = +4 \text{ D}\)

Since both lenses are convex, their powers are positive, which is consistent with our calculations.

Power of Lenses in Contact

When two thin lenses with powers \(P_1\) and \(P_2\) are placed in contact, the net power of the combination \(P_{net}\) is simply the sum of their individual powers. This additive property makes calculating the combined power straightforward.

The formula for the net power of lenses in contact is:

Using the individual powers we calculated:

The net power of the combination of these two convex lenses is +6 dioptres.

Final Result and Option Analysis

The calculated net power of the combination is +6 dioptres. We compare this result with the given options:

• Option 1: +2 dioptre
• Option 2: +6 dioptre
• Option 3: -6 dioptre
• Option 4: +3 dioptre

Our calculated net power (+6 D) matches Option 2.

Revision Table: Key Optics Formulas

Additional Information: Lens Combinations and Types

Understanding lens combinations is crucial in designing optical instruments like telescopes and microscopes. The type of lens (convex or concave) significantly affects the power and the nature of the image formed.

• Convex Lens: Also known as a converging lens. It converges parallel light rays to a real focus. Convex lenses have positive focal length and positive power. They are thicker in the middle than at the edges.
• Concave Lens: Also known as a diverging lens. It diverges parallel light rays so that they appear to originate from a virtual focus. Concave lenses have negative focal length and negative power. They are thinner in the middle than at the edges.
• Combining Lenses:
  • When convex lenses are combined, the total power increases, and the equivalent focal length decreases. The combination acts as a stronger convex lens.
  • When concave lenses are combined, the total power becomes more negative (its magnitude increases), and the equivalent focal length also becomes negative with a smaller magnitude. The combination acts as a stronger concave lens.
  • When a convex and a concave lens are combined, the resulting power can be positive, negative, or zero, depending on the individual powers. This is used in achromatic doublets to reduce chromatic aberration.
• Lenses Not in Contact: If lenses are separated by a distance \(d\), the formula for equivalent focal length is \(\frac{1}{F_{eq}} = \frac{1}{f_1} + \frac{1}{f_2} - \frac{d}{f_1 f_2}\). The net power is \(P_{net} = P_1 + P_2 - d P_1 P_2\).

In this specific problem, since both are convex lenses and are in contact, their positive powers add up, resulting in a combination with greater positive power, which means a shorter equivalent focal length and stronger converging ability.