Problem 59 - Master Resource Book in JEE Main Physics

Question

For light diverging from a point source [NCERT Exemplar] (a) the wavefront is spherical (b) the intensity decreases in proportion to the distance squared (c) the wavefront is parabolic (d) the intensity at the wavefront does not depend on the distance

Step-by-Step Solution

Answer

(a) the wavefront is spherical; (b) the intensity decreases in proportion to the distance squared are correct.

Step 1: Understanding the Question

We need to determine the characteristics of light diverging from a point source. A point source of light emits spherical wavefronts, which means the light spreads out equally in all directions.

Step 2: Analyzing Option (a)

The statement in option (a) is 'the wavefront is spherical'. For a point source of light, the wavefronts are indeed spherical because light emanates in all directions, forming expanding spheres. Thus, option (a) is correct.

Step 3: Analyzing Option (b)

Option (b) states 'the intensity decreases in proportion to the distance squared'. This corresponds to the inverse square law, which applies to point sources. As the wavefronts expand, the energy spreads over a larger area, and the intensity decreases with the square of the distance. Hence, option (b) is correct.

Step 4: Analyzing Option (c)

Option (c) claims 'the wavefront is parabolic'. However, parabolic wavefronts are typically associated with waves in specific setups, not for a point source which naturally forms spherical wavefronts. Therefore, option (c) is incorrect.

Step 5: Analyzing Option (d)

In option (d), 'the intensity at the wavefront does not depend on the distance' is stated. Since intensity decreases with the square of the distance from a point source (as per the inverse square law), this statement is incorrect. Option (d) is therefore not correct.

Key Concepts

Spherical Wavefronts

When light diverges from a point source, it travels outward in all directions, forming spherical wavefronts. Imagine dropping a pebble in a still pond; the ripples spread out in circles. Similarly, in three dimensions, the light expands in spherical fronts. Each wavefront is an imaginary surface that represents the crest of the wave at a particular point in time. These wavefronts expand outward as they move further from the source.

Spherical wavefronts occur naturally with point sources because light emanates equally in all directions. This isotropy is why the wavefronts maintain their spherical shape as they travel. The concept of wavefronts is crucial to understanding complex optical phenomena, as it simplifies the representation of wave propagation over time.

Inverse Square Law

The inverse square law describes how the intensity of light diminishes as it travels away from a point source. As light spreads out, its energy is distributed over a larger area. Mathematically, this is expressed as:\[ I = \frac{P}{4\pi r^2} \]where:

• \(I\) is the intensity of light
• \(P\) is the power of the point source
• \(r\) is the distance from the source
• \(4\pi r^2\) represents the surface area of the spherical wavefront

As the distance \(r\) increases, the intensity \(I\) decreases, since the same amount of light energy is spread over a larger sphere. This fundamental principle helps us understand why light appears dimmer from a distance and plays an essential role in various fields of science, including astronomy and physics.

Point Source of Light

A point source of light is a theoretical concept in physics where light is emitted from a singular point in space without any dimensions. In reality, perfect point sources do not exist, but many sources can be approximated as point sources if their size is negligible compared to the distances involved. This simplification is handy when analyzing problems involving light and optics.

Point sources are pivotal because they help us model and predict how light behaves as it moves through different media. They are the starting place for understanding wave propagation and are used extensively in scientific studies to simplify complex optical systems into more manageable calculations.