Problem 74 - College Physics

Question

A lens made of fused quartz \((n=1.46)\) has a focal length of \(+45 \mathrm{~cm}\) when it is in air. (a) If the lens is immersed in oil \((n=1.50)\), the lens will (1) remain converging, (2) become diverging, (3) have an infinite focal length. Explain. (b) What is the focal length when it is in oil?

Step-by-Step Solution

Answer

(a) It becomes diverging (2). (b) -43.8 cm.

Step 1: Understanding focal length in different mediums

When a lens is immersed in a medium other than air, its focal length changes based on the refractive indices of the lens material and the surrounding medium. The lens formula involving refractive indices is: \[ \frac{1}{f_{medium}} = \left(\frac{n_{lens}}{n_{surrounding}} - 1\right) \left(\frac{1}{R_1} - \frac{1}{R_2}\right) \] where \(f_{medium}\) is the focal length in the medium, \(n_{lens}\) is the refractive index of the lens, and \(n_{surrounding}\) is the refractive index of the medium.

Step 2: Determine the nature of the lens in oil

Given \(n_{lens} = 1.46\) for quartz and \(n_{oil} = 1.50\), we compare these values. Since the refractive index of the oil is greater than that of the lens material, the lens will no longer converge light (become converging). Instead, it will act as a diverging lens, option (2).

Step 3: Calculating the focal length in oil

Substitute the relevant values into the lens formula. Using the given air focal length \( f_{air} = 45\, \text{cm} \), the effective focal length in oil can be approximated by the ratio of refractive indices: \[ f_{oil} = \frac{n_{lens}}{n_{oil}} \times f_{air} = \frac{1.46}{1.50} \times 45 \text{ cm} \approx 43.8 \text{ cm}\] But because it becomes a diverging lens, the actual focal length magnitude will be negative in nature.

Key Concepts

Focal Length

The focal length of a lens determines where it focuses light. It is usually measured in centimeters or meters. Positive focal lengths indicate converging lenses which bring parallel light rays together, ideally forming an image at a specific point. Negative focal lengths, on the other hand, indicate diverging lenses that spread parallel light rays apart, making them appear to come from a different point behind the lens. When a lens is placed in a different medium, like from air to oil, its focal length changes. This happens because the bending of light (refraction) is affected by the material the lens is made of and the material it is immersed in. Understanding these changes is crucial for applications like designing lenses for photography, eyeglasses, and optical instruments.

Refractive Index

The refractive index, often denoted as "n", is a number that describes how fast light travels through a material. Materials with a higher refractive index, like oil in our exercise, slow down light more than those with a lower refractive index, like air or quartz. Knowing the refractive index is important for understanding how much light will bend when entering or leaving a medium. Generally, when light moves from a medium with a lower index to a higher one, it bends towards the normal line (an imaginary line perpendicular to the surface). Conversely, light bends away from the normal when moving from a higher index to a lower one. In the context of the exercise, the lens is made of quartz with a refractive index of 1.46 and is immersed in oil with a refractive index of 1.50. The light's bending behavior changes, causing the lens to act differently than when it is in air. This is why the lens shifts from being converging to diverging.

Lens Formula

The lens formula is a mathematical expression that helps calculate a lens's focal length in different mediums. This formula takes into account both the refractive index of the lens and the medium surrounding it. The general form of the lens formula when refractive indices are involved is: \[ \frac{1}{f_{medium}} = \left(\frac{n_{lens}}{n_{surrounding}} - 1\right) \left(\frac{1}{R_1} - \frac{1}{R_2}\right) \]- \(f_{medium}\) is the focal length within a particular medium.- \(n_{lens}\) is the refractive index of the lens material.- \(n_{surrounding}\) is the refractive index of the surrounding medium.This formula shows that the focal length in a new medium depends on the ratio of the refractive indices and any changes to the curvature of the lens surfaces, indicated by \(R_1\) and \(R_2\). In the exercise, the lens had a focal length of 45 cm in air, and when immersed in oil, this value changed because the medium's refractive index was different. Using the ratio of refractive indices, the focal length was recalculated to be approximately 43.8 cm, but with a negative sign indicating its new diverging nature.