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# Two lenses of power -15D and +5D are in contact with each other. The focal length of the combination is:A.-20 cmB.-10 cmC.+20 cmD.+10 cm

Last updated date: 15th Jul 2024
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Hint: We know that while combining lenses, power is an additive quantity. Also, recall that power of a lens is quantitatively the inverse of the lens’ focal length. Using the previous statements, determine the appropriate equivalent focal length of the combination of the two lenses.
Formula Used:
Resultant Power $P =P_1 +P_2$
Power $P = \dfrac{1}{f}$

We are given that we have two lenses with powers $P_1 = -15D$ and $P_2=+5D$ that are placed in contact with each other.
Recall that the power of a lens is a measure of the extent to which a lens is able to bend light in order to magnify or minify the resultant light incident on it. The greater the power of the lens, the greater is its ability to bend or refract light rays passing through it.
When two lenses are placed in contact with each other, the material of the lenses act like they are combined and have a cumulative refractive effect. Note that this effect is valid only in the case of thin lenses close together such that their combined thickness can be ignored. This means that the power of a lens is an approximately additive quantity.
Thus, when the two lenses are placed in contact with each other, their resultant power will be the sum of their individual powers, i.e.,
$P = P_1+P_2$
However, we know that for a lens, as the focal length decreases, the extent to which the light bends increases. This means that the power of a lens is inversely proportional to the focal length of the lens. The above equation can thus be rewritten as:
$\dfrac{1}{f} = \dfrac{1}{f_1}+\dfrac{1}{f_2}$, where f is the effective focal length of the combined lens setup, and $f_1$ and $f_2$ are their individual focal lengths.
$\dfrac{1}{f} = P_1+P_2$
Plugging in the values we get:
$\dfrac{1}{f} = -15+5 = -10$
$\Rightarrow f = -\dfrac{1}{10} = -0.1\;m = -10\;cm$
Thus, the focal length of the combination of the two lenses is found to be $-10\;cm$.

The correct choice would thus be B. -10 cm.

Note:
By convention, the power of a convex lens is positive (since it is a converging lens and converges light to a point), and the power of a concave lens is negative (since it is a diverging lens and disperses light away after refraction). Thus, power is a measure of the converging ability of a convex lens and the diverging ability of a concave lens.
Also, the additive property of the power of lenses can be used to design lens combos to minimise certain defects in images produced by a single lens. Such a lens system, consisting of several lenses, in contact, is commonly used in the design of the objectives of microscopes and telescopes.