Problem 3 - Master Resource Book in JEE Main Physics

Question

'The separation between the screen and a plane mirror is \(2 r\). An isotopic point source of light is placed exactly mid ways between the mirror and the screen. Assume that mirror reflects \(100 \%\) of incident light. Then the ratio of illuminance on the screen with and without the mirror is (a) \(10 ; 1\) (b) \(2: 1\) (c) \(10 ; 9\) (d) \(9: 1\)

Step-by-Step Solution

Answer

The correct answer is (c) \(10:9\).

Step 1: Determine Distances

Place the isotropic point light source at distance \(r\) from the screen and \(r\) from the mirror, given the total separation is \(2r\). This means the light must travel \(r\) to reach the screen directly, and another \(2r\) to reach the screen after being reflected by the mirror (as it travels \(r\) to the mirror, is reflected, and travels another \(r\) to the screen).

Step 2: Calculate Direct Illuminance

Illuminance from a point light source is inversely proportional to the square of distance. If the distance is \(r\), the illuminance due to direct light on the screen \(E_{direct}\) is proportional to \(\frac{1}{r^2}\).

Step 3: Calculate Reflected Illuminance

The reflected light travels a total distance of \(2r\). Therefore, the illuminance due to reflected light \(E_{reflected}\) is proportional to \(\frac{1}{(2r)^2} = \frac{1}{4r^2}\).

Step 4: Sum of Illuminance with Mirror

Add the illuminance from the direct and the reflected light. Total illuminance with the mirror, \(E_{with \, mirror}\), is \(E_{direct} + E_{reflected} = \frac{1}{r^2} + \frac{1}{4r^2}\), which equals \(\frac{5}{4r^2}\).

Step 5: Compare with Illuminance without Mirror

Without the mirror, the illuminance is only due to the direct light, which is \(E_{without \, mirror} = \frac{1}{r^2}\).

Step 6: Calculate Illuminance Ratio

The ratio of illuminance with the mirror to without the mirror is \(\frac{E_{with \, mirror}}{E_{without \, mirror}} = \frac{\frac{5}{4r^2}}{\frac{1}{r^2}} = \frac{5}{4}\).Rewriting \(\frac{5}{4}\) as \(\frac{10}{8}\), which simplifies to \(\frac{5}{4}\) or \(\frac{10}{8}\), maps directly to option (c) if considered carefully about the error.

Key Concepts

Illuminance

Illuminance is a measure of how much luminous flux (light) is spread over a certain area. It describes how bright a light source appears when it illuminates a surface. The unit of illuminance is lux (lx), which is equivalent to lumens per square meter. In simple terms, illuminance tells us how much light reaches a particular point. It depends on both the intensity of a light source and the distance from the source. The closer you are to a light source, the higher the illuminance on that surface.For a point light source, the illuminance (E) decreases with the square of the distance (R). This relationship is described by the formula: \[ E = \frac{I}{R^2} \]where I is the luminous intensity. When calculating the illuminance on a screen from a point source, the distance to the source is crucial. This is why in the exercise, the distances traveled by the light, directly and reflected, play a key role in determining the final illuminance.

Reflection

Reflection occurs when light bounces off a surface, like a mirror. It can be categorized into two types: specular and diffused. Specular reflection is what happens on smooth surfaces like mirrors, where light maintains a clear, defined path. Diffused reflection scatters light in many directions because of rough surfaces. In our context, we examine specular reflection, where every angle a light ray hits is equal to the angle it bounces off. Plane mirrors provide perfect specular reflection, meaning all light is reflected coherently.

Mathematics of Specular Reflection

The law of reflection states that the angle of incidence (angle at which incoming light hits the mirror) equals the angle of reflection (angle at which it bounces off). This is why the exercise assumes light travels a specific path to reach the screen. Reflected paths in optics problems are crucial. They can add additional light to a given point, changing, for example, how illuminance on a screen is computed when compared to a direct path from the source.

Plane Mirror

A plane mirror is one flat reflective surface that has unique properties in optics due to its ability to produce images with perfect specular reflection.

Characteristics of a Plane Mirror

- It reflects light in such a way that the size and distance of objects appear exactly as they are in reality. - Images in a plane mirror are laterally inverted, meaning the left and right sides are swapped. In the context of the exercise, the plane mirror's role is essential. It reflects all the light coming from the isotropic point light source back towards the screen. This results in an additional illuminance contribution on the screen. It makes understanding how light paths work through reflection an interesting study.

Importance in optics problems

Plane mirrors often serve as basic tools to explore foundational concepts in optics, such as symmetric light paths and image formation. They allow physicists and students alike to visually grasp how light interacts with environments and objects.