NEET Practice Questions (MCQs) with Answers & Solutions

According to NCERT, 'Applying the Cartesian sign convention, OM = –u, MI = +v, MC = +R'. This implies that distances measured in the direction of incident light are taken as positive, while those measured in the opposite direction are negative. All distances are measured from the pole/optic centre of the mirror/lens on the principal axis. This is consistent with the standard Cartesian sign convention for optics, as also stated in Section 9.2.1 and Point 3 of the Summary in NCERT (page 247). The given formula $ \frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R} $ is derived using this sign convention.

The NCERT text states, 'Now, by Snell’s law $n_1 \sin i = n_2 \sin r$. Or for small angles $n_1 i = n_2 r$'. This is a direct application of Snell's Law under the small angle approximation, which is crucial for deriving the spherical refraction formula. The small angle approximation allows $\sin \theta \approx \theta$ (in radians).

Given $n_1 = 1$, $n_2 = 1.5$, $u = -100$ cm (object placed in front, so opposite to incident light direction), $R = +20$ cm (convex surface, so center of curvature is in the direction of incident light). Using the formula $ \frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R} $, we have: $ \frac{1.5}{v} - \frac{1}{-100} = \frac{1.5 - 1}{+20} $ which simplifies to $ \frac{1.5}{v} + \frac{1}{100} = \frac{0.5}{20} = \frac{1}{40} $. So, $ \frac{1.5}{v} = \frac{1}{40} - \frac{1}{100} = \frac{5 - 2}{200} = \frac{3}{200} $. Therefore, $v = \frac{1.5 \times 200}{3} = \frac{300}{3} = +100$ cm. The positive sign indicates a real image formed on the right side (in the direction of incident light). This matches Example 9.5 from the NCERT text.

The NCERT text states, 'As before, we take the aperture (or the lateral size) of the surface to be small compared to other distances involved, so that small angle approximation can be made. In particular, NM will be taken to be nearly equal to the length of the perpendicular from the point N on the principal axis.' This small aperture/small angle approximation (also known as the paraxial approximation) is fundamental to simplifying trigonometric relations (like $\sin \theta \approx \theta$ and $\tan \theta \approx \theta$) to derive the linear formula for image formation.

From NCERT, the derivation in Section 9.5.1 states: 'Now, for $\triangle NOC$, $i$ is the exterior angle. Therefore, $i = \angle NOM + \angle NCM$'. And for 'Similarly, $r = \angle NCM – \angle NIM$'. These relations are derived using basic geometric properties of triangles and are essential steps before applying Snell's Law and small angle approximations.

Though not explicitly stated in the context for refraction at spherical surfaces, the context for wave optics (Equations 10.1 to 10.7) implicitly shows that wavelength and speed change upon refraction ($v_1/v_2 = \lambda_1/\lambda_2$). The frequency of light is an intrinsic property of the source and does not change when light travels from one medium to another. Wavelength and speed change because the optical density of the medium changes. Therefore, frequency remains constant.

Point 3 of the Summary on page 247 states: 'Cartesian sign convention: Distances measured in the same direction as the incident light are positive; those measured in the opposite direction are negative. All distances are measured from the pole/optic centre of the mirror/lens on the principal axis.' This convention is standard for deriving and applying optical formulas.

The NCERT text explicitly states regarding this formula: 'It holds for any curved spherical surface.' (Page 233, after presenting Equation 9.16). This formula is foundational for understanding refraction at a single spherical interface before moving on to lens maker's formula which uses two such surfaces.

As described in NCERT Section 10.3.1 (Wave Optics) and depicted in Figure 10.7(b): 'In Fig. 10.7(b) we consider a plane wave incident on a thin convex lens; the central part of the incident plane wave traverses the thickest portion of the lens and is delayed the most. The emerging wavefront has a depression at the centre and therefore the wavefront becomes spherical and converges to the point F which is known as the focus.' This illustrates the application of Huygens' principle to lens behavior.

From the NCERT text, 'The rays are incident from a medium of refractive index $n_1$, to another of refractive index $n_2$.' (Section 9.5.1). Therefore, $n_1$ is the refractive index of the medium of incidence.