Problem 90 - Physics for Scientists and Engineers with Modern Physics

Question

(a) Show that if two thin lenses of focal lengths \(f_{1}\) and \(f_{2}\) are placed in contact with each other, the focal length of the combination is given by \(f_{\mathrm{T}}=f_{1} f_{2} /\left(f_{1}+f_{2}\right)\). (b) Show that the power \(P\) of the combination of two lenses is the sum of their separate powers, \(P=P_{1}+P_{2}\)

Step-by-Step Solution

Answer

(a) \(f_T = \frac{f_1 f_2}{f_1 + f_2}\); (b) \(P = P_1 + P_2\).

Step 1: Understand lens formula

The lens formula is given by \(\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\), where \(f\) is the focal length, \(d_o\) is the object distance, and \(d_i\) is the image distance. In the context of the problem, we deal with focal lengths.

Step 2: Apply lens formula to each lens

For the first lens of focal length \(f_1\), we have \(\frac{1}{f_1} = \frac{1}{d_o} + \frac{1}{d_1}\). For the second lens of focal length \(f_2\), since the image from the first lens becomes the object for the second lens, we have \(\frac{1}{f_2} = \frac{1}{d_1} + \frac{1}{d_i}\).

Step 3: Combine lens equations

Sum the equations for the two lenses: \(\frac{1}{f_1} + \frac{1}{f_2} = \frac{1}{d_o} + \frac{1}{d_i}\). This represents the system as a whole.

Step 4: Derive formula for combined focal length

Write \(\frac{1}{f_T} = \frac{1}{d_o} + \frac{1}{d_i}\) for the combined system. So, we have \(\frac{1}{f_T} = \frac{1}{f_1} + \frac{1}{f_2}\). Rearrange to find the focal length of the combination: \(f_T = \frac{f_1 f_2}{f_1 + f_2}\).

Step 5: Define lens power

The power \(P\) of a lens is defined as \(P = \frac{1}{f}\). Thus, the power of the individual lenses is \(P_1 = \frac{1}{f_1}\) and \(P_2 = \frac{1}{f_2}\).

Step 6: Calculate combined power

For the combined lenses, \(P_T = \frac{1}{f_T} = \frac{1}{f_1} + \frac{1}{f_2}\). Hence, \(P_T = P_1 + P_2\).

Key Concepts

Thin Lenses

Thin lenses are used extensively in optics. A lens is termed "thin" if its thickness is small compared to its focal length and the radii of curvature of its surfaces. This assumption simplifies calculations significantly, making the thin lens approximation a powerful tool in optical design.
Thin lenses can be either converging (convex) or diverging (concave):

• Converging (Convex) Lenses: These lenses are thicker in the middle and thinner at the edges. They converge parallel rays of light to a focal point.
• Diverging (Concave) Lenses: These lenses are thinner in the middle and thicker at the edges. They spread out parallel rays of light away from a focal point.

The thin lens formula is integral to understand these lenses. It states that \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \), relating the focal length \(f\), object distance \(d_o\), and image distance \(d_i\). This formula helps in determining how an image is formed by the lens. The formula simplifies when lenses are placed in contact, allowing their combined optical power to be calculated accurately.

Focal Length

Focal length is a fundamental concept in lens optics. It is the distance from the lens to the point where parallel rays of light converge or appear to diverge from. The focal length determines how a lens focuses light:

• Positive Focal Length: Indicates a converging lens. Such lenses bring light rays together to focus on a point.
• Negative Focal Length: Indicates a diverging lens. These lenses cause light rays to spread out, as if originating from a focal point behind the lens.

In our exercise, the concept of focal length is used to derive the focal length of two thin lenses in contact. When two lenses of focal lengths \(f_1\) and \(f_2\) are combined, their effective focal length \(f_T\) is given by:\[ f_T = \frac{f_1 f_2}{f_1 + f_2}\] This equation arises from the superposition of their powers, showing how they work together to focus light.

Lens Power

Lens power is a measure of the ability of a lens to bend light, defined as the reciprocal of the focal length. It is expressed in diopters (D), and can be calculated as:\( P = \frac{1}{f} \), where \(f\) is the focal length in meters. High power implies a shorter focal length and stronger focusing ability.
Understanding lens power is vital for optical systems:

• A converging lens has positive power and brings light rays together.
• A diverging lens has negative power and causes light rays to spread apart.

When two lenses are in contact, their combined power \(P_T\) can be calculated by simply adding their powers:\[ P_T = P_1 + P_2\]This shows that the total power of the lens system is the sum of the individual powers, simplifying calculations when designing optical devices. This additive nature helps opticians quickly determine the needed combination of lenses.