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Chapter 31: Problem 64

A double-convex lens with equal 38 -cm curvature radii is made from glass with refractive indices \(n_{\mathrm{red}}=1.51\) and \(n_{\text {violet }}=1.54\) If a point source of white light is on the lens axis \(95 \mathrm{cm}\) from the lens, over what range will its visible image be smeared?

Short Answer

Expert verified
The range over which the visible image will be smeared can be calculated from the difference in image distances for the red and violet light using the object distance, refractive indices and the radius of curvature of the lens.

Step by step solution

01

Find the focal length of red light

Use lensmaker's formula to find the focal length for red light. The lensmaker's formula is given by \[ \frac{1}{f} = (n-1) \left(\frac{1}{R1} - \frac{1}{R2}\right) \] where f is the focal length, n is the refractive index, and R1 and R2 are the radii of curvature of the lens surfaces. Since the lens given is a double-convex lens with equal curvature radii, R1 = R2 = 38cm, and we use the refractive index of red light \(n_{\mathrm{red}}=1.51\). Substituting these values into the lensmaker's formula, we can solve for the focal length for red light, \(f_{\mathrm{red}}\).
02

Find the focal length of violet light

Repeat the above step using the refractive index of violet light, \(n_{\text {violet }}=1.54\), to find the focal length for violet light, \(f_{\mathrm{violet}}\).
03

Find the image distance for red light

Once we have the focal length for red light, we can use the thin lens formula, \( \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \), to find the distance \(v_{\mathrm{red}}\) of the red image from the lens, where u is the object distance from the lens (given as 95 cm) and v is the image distance from the lens.
04

Find the image distance for violet light

Repeat the above step using the focal length for the violet light to find the distance \(v_{\mathrm{violet}}\) of the violet image from the lens.
05

Calculate the range over which the image is smeared

From the image distances calculated in the previous two steps, find the difference between them to calculate the range over which the image is smeared, which is \(|\ v_{\mathrm{red}} - v_{\mathrm{violet}}\ |\).

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