Question

Two converging lenses are in contact. If the focal lengths are each \(5 \mathrm{~cm}\), what is the equivalent focal length of the combination? 1\. \(0.1 \mathrm{~cm}\) 2\. \(2.5 \mathrm{~cm}\) 3\. \(5.0 \mathrm{~cm}\) 4\. \(10.0 \mathrm{~cm}\)

Short answer

\(f_{eq} = 2.5 \mathrm{~cm}\)

Step-by-step solution

Lens formula

The lens maker's formula relates the focal length of a lens to its optical power, which operates according to the principle of superposition when multiple lenses are in contact. The equation for lens formula is: \( \frac{1}{f_{eq}} = \frac{1}{f_1} + \frac{1}{f_2} \) Where \(f_{eq}\) is the equivalent focal length, and \(f_1\) and \(f_2\) are the individual focal lengths of the lenses in contact.

Applying the lens formula

For this problem, the focal lengths of both lenses are given as 5 cm. We can thus substitute this into our lens formula: \( \frac{1}{f_{eq}} = \frac{1}{5} + \frac{1}{5} \)

Solve for equivalent focal length

Now, we can add the two fractions on the right side of the equation and solve for the equivalent focal length, \(f_{eq}\): \( \frac{1}{f_{eq}} = \frac{2}{5} \) To find the value of \(f_{eq}\), we can take the reciprocal of both sides: \( f_{eq} = \frac{5}{2} \)

Finding the answer

This gives us an equivalent focal length of \(f_{eq} = 2.5 \mathrm{~cm}\). Therefore, the correct answer is: 2. 2.5 cm