Question

What would you expect to happen to the magnitude of the power of a lens when it is placed in water $(n=1.33)?$ a) It would increase. d) It would depend if the b) It would decrease. lens was converging or c) It would stay the same. diverging.

Short answer

Answer: It would decrease.

Step-by-step solution

Recall the Lensmaker's Equation

The Lensmaker's Equation relates the focal length (f) of a lens to its refractive indices (n_l and n_m) and its radii of curvature (R_1 and R_2), with n_m being the index of the surrounding medium: $f=\frac{n_l-n_m}{(n_l/R_1)-(n_m/R_2)}$

Determine the power of a lens

The power (P) of a lens is the inverse of its focal length (f): $P=\frac{1}{f}$

Analyze how the surrounding medium affects the focal length

From the Lensmaker's Equation, we can see that when the refractive index of the surrounding medium (n_m) increases, the difference (n_l - n_m) decreases, causing the focal length (f) to increase.

Determine how the focal length affects the power of the lens

When the focal length (f) increases, the power (P) of the lens, according to the power equation, decreases: $P=\frac{1}{f}$ Therefore, the magnitude of the power of the lens would decrease when placed in water.

Choose the correct answer

Based on our analysis in the previous steps, we can now choose the correct option among a), b), c), and d). The correct answer is: b) It would decrease.

Key concepts

Lensmaker's Equation

Understanding the Lensmaker's Equation is essential for analyzing how lenses focus light. This equation is a formula used to calculate the focal length of a lens in air, given its shape and the refractive index of the material from which the lens is made. Mathematically expressed as:

$$f=\frac{n_l-n_m}{\left(\frac{n_l}{R_1}\right)-\left(\frac{n_m}{R_2}\right)}$$

Here, $f$ is the focal length of the lens, $n_l$ is the refractive index of the lens material, and $n_m$ is the refractive index of the medium surrounding the lens. $R_1$ and $R_2$ represent the radii of curvature of the lens's two surfaces. When a lens is placed in a different medium, such as water, the refractive index of the surrounding medium $n_m$ changes, impacting the focal length calculated by this equation.

• Lens shape and material dictate its focusing ability.
• Changing the medium modifies the lens' focal length.

Refractive Index

The refractive index, symbolized as $n$, is a dimensionless number that describes how fast light travels through a material. It is defined as the ratio of the speed of light in a vacuum to the speed of light in the given material. The higher the refractive index, the more the light is slowed down and bent, or refracted, when entering the material.

This property not only affects how lenses bend light to focus or diverge it but also determines how much the light bends when moving from one medium to another. It's crucial when considering how lenses will behave when submerged in different mediums with varying refractive indices. For example, water has a refractive index of approximately $1.33$, which causes light to slow down and bend differently compared to air, which has a refractive index of approximately $1$. This change affects the power of lenses, as shown in the Lensmaker's Equation.

• Higher refractive index means greater bending of light.
• Each material's unique