Question

An image of sun is formed by a lens of focal length \(30 \mathrm{~cm}\) on the metal surface of a photo-electric cell and a photoelectric current (I) is produced. The lens forming the image is then replaced by another of the same diameter but of focal length of \(15 \mathrm{~cm}\). The photoelectric current in this case is \(\ldots \ldots \ldots\) (A) \((1 / 2)\) (B) 1 (C) \(2 \mathrm{I}\) (D) \(4 \mathrm{I}\)

Short answer

The photoelectric current in the second case is \(\frac{1}{4} I\), so the correct answer is (D) \(4 \mathrm{I}\).

Step-by-step solution

Determine the image distance for both lenses 1 and 2

To get the image distance for each lens, we need to remember that the Sun is considered to be at an infinite distance and for such an object, the image forms at the focal point. So, for lens 1 of focal length \(30\text{cm}\), the image distance is \(30\text{cm}\). For lens 2 of focal length \(15\text{cm}\), the image distance is \(15\text{cm}\).

Calculate the light intensity ratio

To calculate the light intensity ratio, we have to apply the inverse square law of light intensity for both lenses. Let the light intensities for lenses 1 and 2 be \(I_1 \text{ and } I_2\) respectively. Then, we have the relation: \[\frac{I_1}{I_2} = \frac{d_2^2}{d_1^2}\] where, \(d_1\) is the image distance for lens 1 (30 cm) and \(d_2\) is the image distance for lens 2 (15 cm). By substituting the values for the image distances, we get: \[\frac{I_1}{I_2} = \frac{15^2}{30^2}\]

Find the light intensity ratio

To find the light intensity ratio, we need to simplify the previous expression: \[\frac{I_1}{I_2} = \frac{(15)^2}{(30)^2} = \frac{1}{2^2}\] thus, \[\frac{I_1}{I_2} = \frac{1}{4}\]

Determine the photoelectric current ratio

We know that the photoelectric current is directly proportional to the light intensity. So, the intensity ratio calculated previously is the same as the photoelectric current ratio. Therefore, the ratio between the photoelectric current for lens 1 denoted as \(I\) and the photoelectric current for lens 2 denoted as \(I'\) is: \[\frac{I'}{I} = \frac{1}{4}\] Rewriting this expression in terms of \(I'\), we get: \[I' = \frac{1}{4} I\] From the given options, we can choose the correct answer as: (D) \(4 \mathrm{I}\)