Question

What will be the focal length of a combination of a convex and a concave lens of the same focal length (placed in contact)?

Hint:

Use \(1/F = 1/f_1 + 1/f_2\) with \(f_1=+f\) and \(f_2=-f\).

Answer 1

Step 1: Write the formula for lenses in contact.
For two thin lenses placed in contact, the equivalent focal length \(F\) is given by
\[ \frac{1}{F}=\frac{1}{f_{1}}+\frac{1}{f_{2}} \]

Step 2: Assign the focal lengths with correct signs.
A convex lens is converging, so its focal length is positive: \(f_{1}=+f\).
A concave lens is diverging, so its focal length is negative: \(f_{2}=-f\).
Both have the same magnitude \(f\).

Step 3: Substitute the values.
\[ \frac{1}{F}=\frac{1}{+f}+\frac{1}{-f}=\frac{1}{f}-\frac{1}{f}=0 \]

Step 4: Interpret the result.
\[ \frac{1}{F}=0\ \Rightarrow\ F=\infty \]
An infinite focal length means the combination has zero power \((P=1/F=0)\); the two lenses exactly neutralise each other and light passes through undeviated (the combination behaves like a plane glass plate).

Result:
\[\boxed{F=\infty\quad(\text{power}=0)}\]