The refractive index of material of a plano-convex lens, if the radius of curvature of convex surface is 10 cm and focal length of lens is 30 cm is
Using Lens Maker's formula for a plano-convex lens: \(\frac{1}{f} = (\mu - 1)\frac{1}{R} ⇒ \frac{1}{30} = (\mu - 1)\frac{1}{10} ⇒ \mu - 1 = \frac{1}{3} ⇒ \mu = \frac{4}{3}\).
Critical angle of light passing from water to air is maximum for
Critical angle is given by \(sintheta_c = frac{1}{mu}\). According to Cauchy's formula, the refractive index \(mu\) increases as wavelength decreases (Yellow > Green > Blue > Indigo in wavelength). Therefore, Yellow has the lowest refractive index and thus the maximum critical angle.
A ray of light is incident normally on one face of a thin prism of refractive index \(mu\) and emerges at an angle (e) from the normal. Angle of prism \(A\) is nearly
For normal incidence on the first face, the angle of refraction \(r_1 = 0\), which implies \(r_2 = A\). Applying Snell's law at the second face for small angles, \(e \approx \mu r_2 = \mu A\), giving \(A \approx \frac{e}{\mu}\).
In the displacement method, a convex lens is placed in between an object and a screen. If magnification in the two positions are \(m_1\) and \(m_2\) (\(m_1 > m_2\)) and the distance between two positions of the lens is x, the focal length of the lens is
In displacement method, \(m_1 = \frac{v_1}{u_1}\) and \(m_2 = \frac{v_2}{u_2} = \frac{u_1}{v_1}\). Since the distance between two positions is \(x = v_1 - u_1\), we obtain \(m_1 - m_2 = \frac{x}{f}\), hence \(f = \frac{x}{m_1-m_2}\).
The refractive index of material of a plano-convex lens, if the radius of curvature of convex surface is \(10\text{ cm}\) and focal length of lens is \(30\text{ cm}\) is
Using lens maker's formula for a plano-convex lens, \(f = \frac{R}{\mu - 1}\). Substituting \(R = 10\text{ cm}\) and \(f = 30\text{ cm}\), we get \(30 = \frac{10}{\mu - 1}\) which gives \(\mu - 1 = \frac{1}{3}\). Therefore, \(\mu = \frac{4}{3}\).
Assertion (A): A lens has two principal focal lengths which may be different in magnitude.
Reason (R): The distance of both principal focus from optical centre of lens depend on the two radii of curvature of the lens. Distance of both principal focus from optical centre a lens are same only if radii of curvature of both sides of lens are same.
Assertion (A) is true if the lens is in different media on either side. Reason (R) is false because for a thin lens in the same medium, magnitudes of focal lengths are always equal, irrespective of the equality of radii of curvature.
Assertion (A): A simple microscope may have different magnification for different persons.
Reason (R): All persons must have the same near point distance of \(25\text{ cm}\).
Assertion (A) is true as magnification depends on individual near point, which varies. Reason (R) is false as the near point varies for different individuals and is not universally \(25\text{ cm}\).
Assertion (A): If an object placed on the optic axis of a lens is illuminated by white light, then image formed will be coloured and not exactly white.
Reason (R): The lens has different focal lengths for different colours.
Assertion (A) is true due to chromatic aberration. Reason (R) is true and correctly explains (A) because the refractive index of lens material varies with wavelength, causing different focal lengths for different colors.
Assertion (A): Paraxial rays are always parallel to the principal axis.
Reason (R): A parallel beam parallel to principal axis converges at the focal point.
Assertion (A) is false; paraxial rays are simply close to the axis, not necessarily parallel. Reason (R) is also false because due to spherical aberration, a real parallel beam does not perfectly converge to a single focal point.
Assertion (A): The image focus (\(2^{\text{nd}}\) focus) and the object focus (\(1^{\text{st}}\) focus) are on the opposite side of the biconvex or biconcave lens.
Reason (R): The radii of curvature of a biconvex lens and biconcave lens are on the opposite side of the lens.
Assertion (A) is true; the two principal focal points are on opposite sides of the lens. Reason (R) is true, describing the geometric arrangement of the centers of curvature. However, (R) does not explain (A).