Question

A convex lens of focal length 30 cm is placed in contact with a concave lens of focal length 20 cm. An object is placed at 20 cm to the left of this lens system. The distance of the image from the lens in cm is ____ .

• A. \( \frac{60}{7} \) cm
• B. 30 cm
• C. 15 cm
• D. 45 cm

Hint:

When dealing with a system of lenses in contact, first find the combined focal length using the formula \( \frac{1}{F_{\text{total}}} = \frac{1}{F_1} + \frac{1}{F_2} \), and then apply the lens formula to find the image distance.

Answer 1

The Correct Option is C

To find the distance of the image from the lens, we need to determine the focal length of the lens system and then apply the lens formula. Let's break down the steps:

1. Calculate the equivalent focal length of the lens system.

The focal lengths given are: \(f_1 = 30 \, \text{cm}\) for the convex lens and \(f_2 = -20 \, \text{cm}\) for the concave lens. The formula for the equivalent focal length \(F\) of two lenses in contact is given by:

\(\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2}\)

Substitute the values:

\(\frac{1}{F} = \frac{1}{30} - \frac{1}{20}\)

\(\frac{1}{F} = \frac{2 - 3}{60} = -\frac{1}{60}\)

Thus, \(F = -60 \, \text{cm}\).

1. Use the lens formula to find the image distance \(v\).

The lens formula is:

\(\frac{1}{f} = \frac{1}{v} - \frac{1}{u}\)

Here, \(f = -60 \, \text{cm}\) (the equivalent focal length), and \(u = -20 \, \text{cm}\) (the object distance, negative because the object is to the left). Substitute into the formula:

\(\frac{1}{-60} = \frac{1}{v} - \frac{1}{-20}\)

Simplify and solve for \(\frac{1}{v}\):

\(\frac{1}{-60} = \frac{1}{v} + \frac{1}{20}\)

\(\frac{1}{v} = \frac{-1}{60} - \frac{1}{20}\)

\(\frac{1}{v} = \frac{-1 - 3}{60} = \frac{-4}{60} = \frac{-1}{15}\)

Thus, \(v = -15 \, \text{cm}\).

1. Interpret the result.

The negative sign indicates that the image is formed on the same side as the object, which is to the left of the lens. Therefore, the distance of the image from the lens is \(15 \, \text{cm}\).

Thus, the correct answer is 15 cm.

Answer 2

When two lenses are in contact, their combined focal length \( F_{\text{total}} \) is given by: \[ \frac{1}{F_{\text{total}}} = \frac{1}{F_1} + \frac{1}{F_2} \] Where: - \( F_1 \) is the focal length of the convex lens, which is \( +30 \, \text{cm} \), - \( F_2 \) is the focal length of the concave lens, which is \( -20 \, \text{cm} \) (since the concave lens has a negative focal length). Substituting the values: \[ \frac{1}{F_{\text{total}}} = \frac{1}{30} + \frac{1}{-20} \] \[ \frac{1}{F_{\text{total}}} = \frac{1}{30} - \frac{1}{20} = \frac{2}{60} - \frac{3}{60} = \frac{-1}{60} \] Thus: \[ F_{\text{total}} = -60 \, \text{cm} \] Now, using the lens formula for the combined lens system: \[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \] Where: - \( f = F_{\text{total}} = -60 \, \text{cm} \) (combined focal length), - \( u = -20 \, \text{cm} \) (object distance, taken as negative for an object placed to the left of the lens), - \( v \) is the image distance, which we need to calculate. Substitute the known values: \[ \frac{1}{-60} = \frac{1}{v} - \frac{1}{-20} \] \[ \frac{1}{-60} = \frac{1}{v} + \frac{1}{20} \] Simplifying: \[ \frac{1}{v} = \frac{1}{-60} - \frac{1}{20} = \frac{-1}{60} - \frac{3}{60} = \frac{-4}{60} \] \[ v = \frac{60}{4} = 15 \, \text{cm} \] Thus, the distance of the image from the lens is 15 cm. Therefore, the correct answer is Option (3).