Question

Two thin lenses of focal length \(\mathrm{f}_{1}\) and \(\mathrm{f}_{2}\) are coaxially placed in contact with each other then the power of combination is (A) \(\left[\left(\mathrm{f}_{1}+\mathrm{f}_{2}\right) / 2\right]\) (B) \(\sqrt{\left(f_{1} / f_{2}\right)}\) (C) $\left[\left(\mathrm{f}_{1} \mathrm{f}_{2}\right) /\left(\mathrm{f}_{1}+\mathrm{f}_{2}\right)\right]$ (D) $\left[\left(\mathrm{f}_{1}+\mathrm{f}_{2}\right) /\left(\mathrm{f}_{1} \mathrm{f}_{2}\right)\right]$

Short answer

The power of the combination of two thin lenses with focal lengths \(f_1\) and \(f_2\) placed coaxially in contact with each other is given by: \[ P_\text{comb} = \frac{1}{f_1}+\frac{1}{f_2} \] Comparing this with the given options, the correct answer is (C) \(\left[\left(f_1 f_2\right) /\left(f_1+f_2\right)\right]\).

Step-by-step solution

Lensmaker's formula for thin lenses

The lensmaker's formula for thin lenses states that the power of a lens (P) is the reciprocal of its focal length (f). Mathematically, this is expressed as follows: \[ P = \frac{1}{f} \]

Lens powers combination

When two thin lenses with powers P1 and P2 are placed in contact, their combined power (P_comb) is given by the sum of their individual powers: \[ P_\text{comb} = P_1 + P_2 \]

Calculate the powers of the individual lenses

Using the lensmakers formula, let's find the power of the first lens with focal length \(f_1\). \[ P_1=\frac{1}{f_1} \] Similarly, for the second lens with focal length \(f_2\). \[ P_2=\frac{1}{f_2} \]

Substitute the powers in the combination formula

Now, we can substitute the values of P1 and P2 in the formula for the power of the combination. \[ P_\text{comb} = \frac{1}{f_1}+\frac{1}{f_2} \]

Identify the correct option

Now, we'll compare the equation we obtained with the options given: (A) \(\left[\left(f_1+f_2\right) / 2\right]\) - Incorrect (B) \(\sqrt{\left(f_1 / f_2\right)}\) - Incorrect (C) \(\left[\left(f_1 f_2\right) /\left(f_1+f_2\right)\right]\) - Correct (D) \(\left[\left(f_1+f_2\right) /\left(f_1f_2\right)\right]\) - Incorrect Therefore, the correct answer is (C) \(\left[\left(f_1 f_2\right) /\left(f_1+f_2\right)\right]\).