Question

A lens of a pair of eyeglasses with index of refraction 1.735 has a power of \(2.794 \mathrm{D}\) in water, with \(n=1.333 .\) What is the power of this lens if it is put in air?

Short answer

Answer: The power of the lens when it is in air is approximately 5.1093 diopters.

Step-by-step solution

Understand the problem

The given data includes the index of refraction for the lens, the index of refraction for water, and the power of the lens when it is in water. We need to find the power of the lens when it is in air.

Find the focal length of the lens in water

To find the focal length of the lens in water, we will use the Lens Power formula: Power of the lens = (n_lens - n_water) / f_lens_in_water We will rearrange the formula to find the focal length of the lens in water, f_lens_in_water: f_lens_in_water = (n_lens - n_water) / Power_in_water Here, n_lens = 1.735, n_water = 1.333, and Power_in_water = 2.794D. Substituting these values into the formula: f_lens_in_water = (1.735 - 1.333) / 2.794 = 0.402 / 2.794 = 0.1438 m

Find the index of refraction for air

Since the lens is being placed in air, we need the index of refraction for air, which is typically approximately 1.000.

Find the power of the lens in air

Now, let's find the power of the lens in air. Recall that the lens power formula is Power of the lens = (n_lens - n) / f_lens Here, we are using the same lens, so its index of refraction (n_lens) remains the same. We simply replace the environmental index of refraction (n) with the index of refraction for air: Power_in_air = (n_lens - n_air) / f_lens_in_water Plugging in the values we found in previous steps, we get: Power_in_air = (1.735 - 1.000) / 0.1438 = 0.735 / 0.1438 = 5.1093D Therefore, the power of the lens when it is put in air is approximately 5.1093 diopters.

Key concepts

Index of Refraction

The index of refraction, denoted by the symbol 'n', is a measure of how much a material can bend light. In technical terms, it indicates the factor by which the speed of light is reduced as it passes through that material compared to its speed in a vacuum. A higher index of refraction means the material is more effective at bending light. For instance, typical values for index of refraction are around 1.000 for air and 1.333 for water.

Different materials affect light differently, and knowing a material's index of refraction is crucial when designing lenses, as it directly influences the lens' ability to focus light. In the context of optical lenses, a higher index of refraction also typically means the lens can be made thinner while achieving the same focusing power, which is desirable for something like eyeglasses.

Optical Power

Optical power, expressed in diopters (D), quantifies the ability of a lens to converge or diverge light. Positive diopters indicate a lens that converges light and is used to correct farsightedness, while negative diopters signal a lens that diverges light, used for correcting nearsightedness. The formula to find the power of a lens is given by the difference in the indices of refraction between the lens material and the surrounding medium, divided by the focal length of the lens (in meters).

When we mention the lens' power in the context of water or air, we are discussing how the lens' behavior changes when submerged in water versus when it is exposed to air. This is because the optical power of a lens depends on the surrounding medium's index of refraction, due to the relationship between the speed of light in the medium and the lens material.

Lens Formula

The lens formula relates the index of refraction of a lens, the index of refraction of the surrounding medium, and the lens' focal length to determine the lens' optical power. Mathematically, the power of the lens P is described by the equation:\[\begin{equation}P = \frac{n_{lens} - n_{medium}}{f_{lens}}\end{equation}\]where \(n_{lens}\) is the index of refraction of the lens material, \(n_{medium}\) is the index of refraction of the medium (e.g., air or water), and \(f_{lens}\) is the lens' focal length. The focal length is the distance over which initially parallel rays of light are brought to a focus after passing through the lens.

Employing this formula allows us to understand how the power of a lens will change when moved from one medium to another simply by adjusting the value of \(n_{medium}\). For example, as in the exercise, a lens used under water versus in air would have significantly different powers due to the differing indices of refraction of these media.