Question

A person with a near-point distance of \(24.0 \mathrm{~cm}\) finds that a magnifying glass gives an angular magnification that is 1.25 times larger when the image of the magnifier is at the near point than when the image is at infinity. What is the focal length of the magnifying glass?

Short answer

A: 6.0 cm.

Step-by-step solution

Recall the lens formula and angular magnification equation

To solve this problem, we will need to use the lens formula: \[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \] Where: - \(f\) is the focal length of the lens - \(u\) is the object distance - \(v\) is the image distance We also need to use the equation for angular magnification (\(M\)) of a magnifying lens: \[ M = 1 + \frac{d}{f} \] where \(d\) is the near-point distance.

Determine the magnification at the near point and infinity

Given that the angular magnification when the image is at the near point is 1.25 times larger than when the image is at infinity, let's denote that: \[ M_{near~point} = 1.25 \times M_{infinity} \]

Find the angular magnification at infinity

When the image is at infinity (\(v = \infty\)), the object distance (\(u = f\)) as all parallel rays will converge at the focus. Plugging these values into the lens formula, we get: \[ \frac{1}{f} = \frac{1}{f} + 0 \] Using the equation for angular magnification, we can find the magnification at infinity: \[ M_{infinity} = 1 + \frac{d}{f} \] where \(d = 24.0 \mathrm{~cm}\).

Calculate the magnification at the near point

Using the relation established in Step 2, we can find the magnification at the near point: \[ M_{near~point} = 1.25 \times M_{infinity} \]

Find the object and image distances at the near point

When the image of the magnifier is at the near point, the image distance \(v = d = 24.0 \mathrm{~cm}\). We can use the lens formula to find the object distance at the near point: \[ \frac{1}{f} = \frac{1}{u_{near~point}} + \frac{1}{d} \]

Connect magnification, focal length and object distance

Now we can connect the magnification at the near point, focal length, and object distance using the equation for angular magnification at the near point: \[ M_{near~point} = 1 + \frac{d}{f_{near~point}} \]

Solve for the focal length of the magnifying glass

We have obtained relationships between \(M_{infinity}, M_{near~point}, f_{near~point}\), and \(d\). Using these relationships, we can solve for the focal length (\(f\)) of the magnifying glass. 1. Replace \(M_{infinity}\) in the equation for \(M_{near~point}\) using the formula for magnification at infinity. 2. Substitute the value of \(M_{near~point}\) found in Step 4. 3. Solve the equation to find the focal length, \(f\). After following all the steps, the focal length of the magnifying glass will be determined.