Combination of Coaxial Thin Convex and Concave Lenses

To determine the nature and focal length of the combination of a coaxial thin convex lens and a thin concave lens placed in contact, we use the effective focal length formula for thin lenses in contact:

$\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2}$

where $F$ is the equivalent focal length of the lens combination, $f_1$ is the focal length of the convex lens, and $f_2$ is the focal length of the concave lens.

Applying the sign convention for focal lengths:

• For a convex lens, the focal length is positive: $f_1 = +|f_{\text{convex}}|$
• For a concave lens, the focal length is negative: $f_2 = -|f_{\text{concave}}|$

Substituting these values into the expression for effective power/focal length:

$\frac{1}{F} = \frac{1}{|f_{\text{convex}}|} - \frac{1}{|f_{\text{concave}}|}$

$\frac{1}{F} = \frac{|f_{\text{concave}}| - |f_{\text{convex}}|}{|f_{\text{convex}}| \cdot |f_{\text{concave}}|}$

Now, let us analyze the behavior of the combination based on the sign of the equivalent focal length $F$:

1. Behavior as a Concave Lens: The system behaves as a net concave (diverging) lens if the equivalent focal length $F$ is negative ($\frac{1}{F} < 0$): $\frac{|f_{\text{concave}}| - |f_{\text{convex}}|}{|f_{\text{convex}}| \cdot |f_{\text{concave}}|} < 0$

   Since the denominator $|f_{\text{convex}}| \cdot |f_{\text{concave}}|$ is always positive: $|f_{\text{concave}}| - |f_{\text{convex}}| < 0 \implies |f_{\text{convex}}| > |f_{\text{concave}}|$

2. Behavior as a Convex Lens: The system behaves as a net convex (converging) lens if the equivalent focal length $F$ is positive ($\frac{1}{F} > 0$): $|f_{\text{concave}}| - |f_{\text{convex}}| > 0 \implies |f_{\text{convex}}| < |f_{\text{concave}}|$

3. Effect of Interchanging Positions: Since addition is commutative ($\frac{1}{f_1} + \frac{1}{f_2} = \frac{1}{f_2} + \frac{1}{f_1}$), interchanging the positions of two thin lenses in contact does not change the effective focal length of the system.

Thus, the combination behaves as a concave lens if $|f_{\text{convex}}| > |f_{\text{concave}}|$.

Correct Option: A