Problem 95 - Master Resource Book in JEE Main Physics

Question

A photocell is illuminated by a small bright source placed \(1 \mathrm{~m}\) away. When the same source of light is placed \(\frac{1}{2} \mathrm{~m}\) away, the number of electrons emitted by photo cathode would (a) increase by a factor of 2 (b) decrease by a factor of 2 (c) increase by a factor of 4 (d) decrease by a factor of 4

Step-by-Step Solution

Answer

(c) increase by a factor of 4.

Step 1: Understand the Relationship between Distance and Intensity

The intensity of light from a point source is inversely proportional to the square of the distance from the source. This means that if the distance is halved, the intensity increases by a factor of four (\(I \propto \frac{1}{d^2}\)).

Step 2: Calculate the Change in Intensity

Initially, the source is at \(1 \text{ meter}\) away, and if moved closer to \(\frac{1}{2} \text{ meter}\), the intensity increases by:\[\left(\frac{1}{\frac{1}{2}}\right)^2 = 4.\]Thus, the light intensity increases by a factor of 4.

Step 3: Relate Intensity to Electron Emission

According to the photoelectric effect, the number of electrons emitted is directly proportional to the light intensity. Therefore, if intensity increases by a factor of 4, the number of emitted electrons also increases by a factor of 4.

Step 4: Conclusion and Answer Selection

Based on the above logical deductions, the correct answer to the question is the option that states the number of electrons increases by a factor of 4. Therefore, option (c) is the right choice.

Key Concepts

Distance and Intensity Relationship

When we talk about the relationship between distance and intensity, we're diving into a fundamental aspect of how light behaves. As light travels from its source, it spreads out, and this spreading affects how intense the light is when it reaches a specific point. A key rule here is the "Inverse Square Law." It states that the intensity (\(I\)) of light is inversely proportional to the square of the distance (\(d\)) from the source. This means:

• If you double the distance from the light source, the intensity becomes four times weaker (as \(I \propto \frac{1}{d^2}\)).
• Conversely, if you halve the distance, the intensity increases by four times.

Understanding this relationship helps explain why stars look dimmer when they are further away and why the Sun feels hotter at midday in summer. It's the same principle applied to the photocell exercise, where moving the light source closer squeezes more intensity into a smaller area.

Light Intensity

Light intensity refers to the power per unit area carried by a wave. In simpler terms, it's about how "strong" or "bright" the light is at a particular location. This concept plays a pivotal role in various scientific phenomenons, including the photoelectric effect. When considering light sources like the sun or a bulb, the light emitted disperses into the surrounding environment. As it travels, the intensity diminishes, becoming less potent the further it reaches. In a practical sense, if you have a lamp at the center of a room, the brightness is strongest near the lamp and weaker as you move away. That's why in the given problem, reducing the distance from the light source to the photocell increases the intensity that hits it.

• This increase in intensity results in more energy being transferred to the material where light is absorbed.
• This extra energy can influence phenomena like electron emission as described by the photoelectric effect.

Each of these points reminds us that intensity is not just about how things appear, but how energy is distributed across distances in physical space.

Electron Emission

Electron emission is an essential concept that emerges when discussing the photoelectric effect, wherein electrons are ejected from a material when it absorbs light energy. This process is what Einstein famously explained, which earned him the Nobel Prize. The photoelectric effect hinges on the fact that light consists of "packets" of energy called photons. Each photon carries energy proportional to its frequency. When these photons hit a material, they can transfer energy to electrons. If an electron absorbs enough energy, it can overcome the forces holding it inside the atom and shoot off into space. Three primary factors influence electron emission:

• Light Intensity: More intense light means more photons hitting the surface, increasing the probability of electron emission.
• Frequency of Light: Higher frequency light carries more energy. As such, even lower intensity light with a high enough frequency can cause electron emission.
• Material Properties: Different materials require different amounts of energy (work function) to release electrons.

In the question, as the light intensity increases by moving the light closer, it boosts the number of photons interacting with the material, thus increasing the number of emitted electrons. This shows the direct relation between light intensity and electron emission in practice.