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Chapter 31: Problem 22

A lens with 50 -cm focal length produces a real image the same size as the object. How far from the lens are image and object?

Short Answer

Expert verified
The object and image are both 100 cm from the lens.

Step by step solution

01

Determine the lens equation

The thin lens equation is \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \).
02

Set up the equation using the given values

We know that the object and image distances are equal (since the image and object are the same size), thus \( d_o = d_i \). Substitute \( d_i = d_o \) into lens equation so it becomes \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_o} \) which simplifies to \( \frac{1}{f} = \frac{2}{d_o} \). The given focal length \( f \) is 50 cm.
03

Solve for Do

Inserting the given value into the equation, we get \( \frac{1}{50} = \frac{2}{d_o} \). Solving this for \( d_o \) gives \( d_o = 2 * 50 \), so \( d_o = 100 \) cm.
04

Solve for Di

As the image and object are at the same distance from the lens, \( d_i \) is also 100 cm.

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Most popular questions from this chapter

Two specks of dirt are trapped in a crystal ball, one at the center and the other halfway to the surface. If you peer into the ball on a line joining the two specks, the outer one appears to be only one-third of the way to the other. Find the refractive index of the ball.

Viewed from Earth, the Moon subtends an angle of \(0.52^{\circ}\) in the sky. What will be the physical size of the Moon's image formed by either of the twin Keck telescopes, with 10 -m-diameter mirrors and 17.5 -m focal length?

Generalize the derivation of the lensmaker's formula (Equation 31.7 ) to show that a lens of refractive index \(n_{\text {lens }}\) in an external medium with index \(n_{\mathrm{ext}}\) has focal length given by $$\frac{1}{f}=\left(\frac{n_{\mathrm{lens}}}{n_{\mathrm{ext}}}-1\right)\left(\frac{1}{R_{1}}-\frac{1}{R_{2}}\right)$$

Show that placing a 1 -diopter lens in front of a 2 -diopter lens gives the equivalent of a single 3 -diopter lens (i.e., the powers of closely spaced lenses add).

An object and its lens-produced real image are \(2.4 \mathrm{m}\) apart. If the lens has \(55-\mathrm{cm}\) focal length, what are the possible values for the object distance and magnification?
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