Where to place object for silvered biconvex lens (R=20cm, μ=1.5) so image coincides?

10 cm. f_lens=20cm, f_mirror=10cm, P_eq=1/5, f_eq=5cm. Object at 2f_eq=10cm.

What is the equivalent focal length?

f_eq=5cm. P_eq=2P_lens+P_mirror=2/20+1/10=1/5.

Silvered biconvex lens both R=20cm μ=1.5 — object distance for coincident image

JEERANKERS

PhysicsOptics

QMCQOptics

A thin biconvex lens ($\mu=1.5$) with equal radii of 20 cm each has its left surface silvered. To have the image and object at the same place, the object should be placed at ________ cm from the lens.

A) 10 B) 12.5 C) 13 D) 13.5

✅ Correct Answer

A) 10 cm

✓

Solution

Focal length of the refracting lens

$\dfrac{1}{f_L}=(\mu-1)\!\left(\dfrac{1}{R_1}-\dfrac{1}{R_2}\right)=0.5\!\left(\dfrac{1}{20}+\dfrac{1}{20}\right)=\dfrac{1}{20}\ \Rightarrow\ f_L=20\text{ cm}$

Focal length of silvered mirror

Left convex surface silvered → acts as concave mirror: $f_m=R/2=10$ cm.

Equivalent mirror power

$P_{eq}=\dfrac{2}{f_L}+\dfrac{1}{f_m}=\dfrac{2}{20}+\dfrac{1}{10}=\dfrac{1}{10}+\dfrac{1}{10}=\dfrac{1}{5}\ \Rightarrow\ f_{eq}=5\text{ cm}$

Object at centre of curvature of equivalent mirror

$$d=2f_{eq}=2\times5=\boxed{10\text{ cm}}$$

📘 Equivalent Mirror for Silvered Lens

Silvered lens acts as equivalent mirror with $P_{eq}=2P_{lens}+P_{mirror}$. For image to coincide with object, object must be at centre of curvature = $2f_{eq}$.

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Important Concepts

P_eq=2P_lens+P_mirrorLight refracts through lens twice (before and after reflection) and reflects once. Powers add for each operation.

f_lens=20cmLensmaker: 1/f=(0.5)(1/20+1/20)=1/20. f=20cm.

f_mirror=10cmSilvered convex surface acts as concave mirror. f_m=R/2=20/2=10cm.

Object at 2f_eq=10cmf_eq=5cm. For image to coincide: object at 2×5=10cm from lens.

FAQs

Formula for equivalent mirror power?

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P_eq=2P_lens+P_mirror. Here: 2×(1/20)+1/10=1/5. f_eq=5cm.

Why multiply P_lens by 2?

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Light passes through the lens TWICE — once going toward the mirror, once coming back after reflection.

What if both surfaces were silvered?

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Not physical — only one silvered surface creates a mirror-lens combination.

What is object at 10cm from lens?

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At 2f_eq=10cm, rays from object reflect back through the same path and form an image at the same point as the object.

Is this from JEE Main 2026?

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Yes, this appeared in JEE Main 2026.

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