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Geometrical Optics question

2022 · 26 Jul · Shift 1 · Q70
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  5. /2022 · 26 Jul · Shift 1 · Q70

Geometrical Optics question

2022 · 26 Jul · Shift 1 · Q70

JEE MainPhysicsGeometrical OpticsNumerical+4 / −1
The graph between 1u\frac{1}{u}u1​ and 1v\frac{1}{v}v1​ for a thin convex lens in order to determine its focal length is plotted as shown in the figure. The refractive index of lens is 1.51.51.5 and its both the surfaces have same radius of curvature RRR. The value of RRR will be ‾\underline{\hspace{2cm}}​cm.\mathrm{cm} .cm.(where u=u=u= object distance, v=v=v= image distance) JEE Main 2022 (Online) 26th July Morning Shift Physics - Geometrical Optics Question 107 English
Numerical answer
View written solutionFree

Correct answer: 10

  1. Lens formula in terms of 1u\frac{1}{u}u1​ and 1v\frac{1}{v}v1​

For a thin lens, using magnitudes,

1f=1u+1v\frac{1}{f}=\frac{1}{u}+\frac{1}{v}f1​=u1​+v1​

So if we plot 1v\frac{1}{v}v1​ versus 1u\frac{1}{u}u1​, the relation is

1v=−1u+1f\frac{1}{v}=-\frac{1}{u}+\frac{1}{f}v1​=−u1​+f1​

which is a straight line of slope −1-1−1 and intercepts on both axes equal to 1f\frac{1}{f}f1​.

  1. Read focal length from the graph

From the given graph, the intercept on the axis is

1f=0.2 cm−1\frac{1}{f}=0.2\ \text{cm}^{-1}f1​=0.2 cm−1

Hence,

f=10.2=5 cmf=\frac{1}{0.2}=5\ \text{cm}f=0.21​=5 cm
  1. Use lens maker's formula

For a thin convex lens in air,

1f=(μ−1)(1R1−1R2)\frac{1}{f}=(\mu-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)f1​=(μ−1)(R1​1​−R2​1​)

Given:

  • refractive index μ=1.5\mu=1.5μ=1.5
  • both surfaces have same radius of curvature RRR

For a symmetric biconvex lens,

R1=+R,R2=−RR_1=+R,\qquad R_2=-RR1​=+R,R2​=−R

So,

1f=(1.5−1)(1R−(−1R))\frac{1}{f}=(1.5-1)\left(\frac{1}{R}-\left(-\frac{1}{R}\right)\right)f1​=(1.5−1)(R1​−(−R1​)) 1f=0.5⋅2R=1R\frac{1}{f}=0.5\cdot \frac{2}{R}=\frac{1}{R}f1​=0.5⋅R2​=R1​

Thus,

R=fR=fR=f

Since f=5 cmf=5\ \text{cm}f=5 cm,

R=5 cmR=5\ \text{cm}R=5 cm
  1. Final answer
5\boxed{5}5​
  1. Comparison with stored answer

Stored correct answer is 101010, but from the graph-intercept method and lens maker formula, the radius comes out to be 5 cm5\,\text{cm}5cm.

So I do not agree with the stored answer.

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