JEE Advanced 2024 · Paper 1 · Physics Q7

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PhysicsMultiple correct+4 / -2Question 24 of 51

A glass beaker has a solid, plano-convex base of refractive index 1.60, as shown in the figure. The radius of curvature of the convex surface (SPU) is $9\,\mathrm{cm}$ , while the planar surface (STU) acts as a mirror. This beaker is filled with a liquid of refractive index $n$ up to the level QPR. If the image of a point object O at a height of $h$ (OT in the figure) is formed onto itself, then, which of the following option(s) is(are) correct?

Cross-section of a beaker with vertical side walls: the left wall runs from Q (upper) down to S (bottom corner), the right wall from R (upper) down to U (bottom corner). The flat bottom surface S-T-U is the planar face of a solid plano-convex glass base (shaded grey); its upper convex surface rises from S and U to a highest point P at the centre. A liquid (dotted shading) fills the space above the convex surface up to the horizontal level Q-P-R, which just touches the top P of the convex surface. A dashed vertical axis passes through the point object O (above the beaker), P and T (the centre of the bottom surface); OT is the height h of the object above the bottom.

For

n = 1.42

h = 50\,\mathrm{cm}

For

n = 1.35

h = 36\,\mathrm{cm}

For

n = 1.45

h = 65\,\mathrm{cm}

For

n = 1.48

h = 85\,\mathrm{cm}

Answer and worked solution

Official answer: A, B

For the image to form on the object, rays must strike the plane mirror STU normally, i.e. travel parallel to the axis inside the glass. Near the axis the liquid layer and the glass both meet at P, so treat the two refractions as happening at P.

Flat liquid surface (air → liquid): the object at height $h$ appears at distance $nh$ above P.

Convex surface (liquid $n$ → glass $1.6$ ), centre of curvature in the glass, $R = +9\,\mathrm{cm}$ (light travelling downward), image at infinity:
$\frac{1.6}{\infty} - \frac{n}{-nh} = \frac{1.6 - n}{9} \implies h = \frac{9}{1.6 - n}\ \mathrm{cm}.$

$n = 1.42$ : $h = 50\,\mathrm{cm}$ ✓
$n = 1.35$ : $h = 36\,\mathrm{cm}$ ✓
$n = 1.45$ : $h = 60\,\mathrm{cm}$ (not 65) ✗
$n = 1.48$ : $h = 75\,\mathrm{cm}$ (not 85) ✗

Topics:Refraction at spherical surfaces Combination of lenses and mirrors

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JEE Advanced 2024 · Paper 1, Physics question 7