Problem Analysis:
- We are given two lenses: a biconvex lens (refractive index $n_b = 1.5$) and a plano-concave lens (refractive index $n_p = 1.7$).
- The magnitudes of their powers are equal: $|P_{biconvex}| = |P_{planoconcave}|$.
- The magnitude of the radius of curvature of the plano-concave lens equals the magnitude of the radius of curvature of the back surface of the biconvex lens. Let this common magnitude be $R$.
- We need to find the ratio of the radius of curvature of the front surface ($R_{b1}$) to the back surface ($R_{b2}$) of the biconvex lens. We'll denote the magnitudes as $R_{b1}'$ and $R_{b2}'$.
Lens Maker's Formula Application
The Lens Maker's formula is given by: $P = (n-1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right)$ where $n$ is the refractive index, $R_1$ is the radius of the first surface, and $R_2$ is the radius of the second surface. Sign convention is crucial.
Biconvex Lens Power
For a biconvex lens with refractive index $n_b = 1.5$: Let the front surface radius be $R_{b1}$ (convex, so positive magnitude $R_{b1}'$). Let the back surface radius be $R_{b2}$ (convex, but facing the second medium, so negative value $-R_{b2}'$, where $R_{b2}'$ is the positive magnitude). The power $P_b$ is: $P_b = (n_b - 1) \left( \frac{1}{R_{b1}'} - \frac{1}{-R_{b2}'} \right) = (1.5 - 1) \left( \frac{1}{R_{b1}'} + \frac{1}{R_{b2}'} \right)$ $P_b = 0.5 \left( \frac{1}{R_{b1}'} + \frac{1}{R_{b2}'} \right)$ The magnitude of the power is $|P_b| = 0.5 \left( \frac{1}{R_{b1}'} + \frac{1}{R_{b2}'} \right)$.
Plano-Concave Lens Power
For a plano-concave lens with refractive index $n_p = 1.7$: The first surface is plane, so $R_{p1} = \infty$. The second surface is concave, so its radius value is negative. Let the magnitude be $R_{p2}'$, thus $R_{p2} = -R_{p2}'$. The power $P_p$ is: $P_p = (n_p - 1) \left( \frac{1}{R_{p1}} - \frac{1}{R_{p2}} \right) = (1.7 - 1) \left( \frac{1}{\infty} - \frac{1}{-R_{p2}'} \right)$ $P_p = 0.7 \left( 0 + \frac{1}{R_{p2}'} \right) = \frac{0.7}{R_{p2}'}$ The magnitude of the power is $|P_p| = \frac{0.7}{R_{p2}'}$.
Calculating the Radius Ratio
We are given that $|P_b| = |P_p|$ and the magnitude of curvature match $|R_{p2}'| = |R_{b2}'| = R$. Substituting these into the power equation:
$0.5 \left( \frac{1}{R_{b1}'} + \frac{1}{R} \right) = \frac{0.7}{R}$
Multiply both sides by $R$ to simplify:
$0.5 \left( \frac{R}{R_{b1}'} + 1 \right) = 0.7$
Isolate the term with $R_{b1}'$:
$\frac{R}{R_{b1}'} + 1 = \frac{0.7}{0.5} = 1.4$
$\frac{R}{R_{b1}'} = 1.4 - 1 = 0.4$
Convert $0.4$ to a fraction:
$\frac{R}{R_{b1}'} = \frac{4}{10} = \frac{2}{5}$
Now, find the ratio $\frac{R_{b1}'}{R}$ by inverting the fraction:
$\frac{R_{b1}'}{R} = \frac{5}{2}$
Since $R = R_{b2}'$, the ratio of the radius of curvature of the front surface to the back surface of the biconvex lens is:
$R_{b1}' : R_{b2}' = 5:2$
Conclusion
The ratio of the radius of curvature of the front and back surface of the biconvex lens is 5:2.
Similar Questions
- Given below are two statements:
**Statement I:** A plane wave after passing through prism remains as plane wave but passing through small pin hole may become spherical wave.
**Statement II:** The curvature of a spherical wave emerging from a slit will increase for increasing slit width.
In the light of the above statements, choose the correct answer from the options given below - A convex lens of refractive index $1.5$ and focal length $f = 18 \text{ cm}$ is immersed in water. The difference in focal lengths of the given lens when it is in water and in air is $\alpha \times f$. The value of $\alpha$ is ________.
(refractive index of water = $4/3$) - Consider an equilateral prism (refractive index $\sqrt{2}$). A ray of light is incident on its one surface at a certain angle $i$. If the emergent ray is found to graze along the other surface then the angle of refraction at the incident surface is close to _________.
- A thin convex lens of focal length 5 cm and a thin concave lens of focal length 4 cm are combined together (without any gap) and this combination has magnification $m_1$ when an object is placed 10 cm before the convex lens. Keeping the positions of convex lens and object undisturbed a gap of 1 cm is introduced between the lenses by moving the concave lens away, which lead to a change in magnification of total lens system to $m_2$. The value of $\left| \frac{m_1}{m_2} \right|$ is _________.
- A parallel beam of light travelling in air (refractive index 1.0) is incident on a convex spherical glass surface of radius of curvature 50 cm. Refractive index of glass is 1.5. The rays converge to a point at a distance x cm from the centre of the curvature of the spherical surface. The value of x is _________ cm.
- Which of the following are true for a single slit diffraction ?
A. Width of central maxima increases with increase in wavelength keeping slit width constant.
B. Width of central maxima increases with decrease in wavelength keeping slit width constant.
C. Width of central maxima increases with decrease in slit width at constant wavelength.
D. Width of central maxima increases with increase in slit width at constant wavelength.
E. Brightness of central maxima increases for decrease in wavelength at constant slit width. - The wavelength of light, while it is passing through water is 540 nm. The refractive index of water is $4/3$. The wavelength of the same light when it is passing through a transparent medium having refractive index of $3/2$ is _________ nm.
- In parallax method for the determination of focal length of a concave mirror, the object should always be placed :
- A laser beam has intensity of $4.0 \times 10^{14} \text{ W/m}^2$. The amplitude of magnetic field associated with beam is _________ T. (Take $\varepsilon_0 = 8.85 \times 10^{-12} \text{ C}^2\text{/Nm}^2$ and $c = 3 \times 10^8 \text{ m/s}$)
- An unpolarised light is incident at an interface of two dielectric media having refractive indices of 2 (incident medium) and $2\sqrt{3}$ (medium) respectively. To satisfy the condition that reflected and refracted rays are perpendicular to each other, the angle of incidence is ______ .
Important Questions from Optics
- Given below are two statements:
**Statement I:** A plane wave after passing through prism remains as plane wave but passing through small pin hole may become spherical wave.
**Statement II:** The curvature of a spherical wave emerging from a slit will increase for increasing slit width.
In the light of the above statements, choose the correct answer from the options given below - A convex lens of refractive index $1.5$ and focal length $f = 18 \text{ cm}$ is immersed in water. The difference in focal lengths of the given lens when it is in water and in air is $\alpha \times f$. The value of $\alpha$ is ________.
(refractive index of water = $4/3$) - Consider an equilateral prism (refractive index $\sqrt{2}$). A ray of light is incident on its one surface at a certain angle $i$. If the emergent ray is found to graze along the other surface then the angle of refraction at the incident surface is close to _________.
- A thin convex lens of focal length 5 cm and a thin concave lens of focal length 4 cm are combined together (without any gap) and this combination has magnification $m_1$ when an object is placed 10 cm before the convex lens. Keeping the positions of convex lens and object undisturbed a gap of 1 cm is introduced between the lenses by moving the concave lens away, which lead to a change in magnification of total lens system to $m_2$. The value of $\left| \frac{m_1}{m_2} \right|$ is _________.
- A parallel beam of light travelling in air (refractive index 1.0) is incident on a convex spherical glass surface of radius of curvature 50 cm. Refractive index of glass is 1.5. The rays converge to a point at a distance x cm from the centre of the curvature of the spherical surface. The value of x is _________ cm.
