Question

An incandescent bulb has a thin filament of tungsten that is heated to high temperature by passing an electric current. The hot filament emits black-body radiation. The filament is observed to break up at randon locations after a sufficiently long time of operation due to non-uniform evaporation of tungsten from the filament. If the bulb is powered at constant voltage, which of the following statement(s) is(are) true? (A) The temperature distribution over the filament is uniform (B) The resistance over small sections of the filament decreases with time (C) The filament emits more light at higher band of frequencies before it breaks up (D) The filament consumes less electrical power towards the end of the life of the bulb

Short answer

Statements (B), (C), and (D) are true. (A) is false because the temperature distribution is non-uniform due to non-uniform evaporation of the filament.

Step-by-step solution

Evaluate Statement A

The temperature distribution over the filament would not be uniform toward the end of the bulb's life due to non-uniform evaporation of tungsten. This would cause variations in the filament thickness, leading to non-uniform resistance and therefore non-uniform temperature.

Evaluate Statement B

As the filament's tungsten evaporates non-uniformly, the thinner sections of the filament would have higher resistance (since resistance increases as the cross-sectional area decreases). However, as the temperature increases, the resistance of metallic conductors like tungsten also increases. Thus, considering long periods of operation and the increasing resistance due to temperature, this statement is true.

Evaluate Statement C

According to Planck's law, the intensity and frequency of black-body radiation increase with temperature. Assuming the temperature of the filament to increase in the areas where tungsten has evaporated, the filament would indeed emit more light at higher frequencies before breaking up. Hence, this statement is true.

Evaluate Statement D

If the resistance of the filament increases over time because it gets thinner, the electrical power consumed (\( P = V^2/R \text{, where } V \text{ is constant voltage and } R \text{ is resistance}\) is calculated using the power formula) would decrease. Therefore, the bulb would consume less electrical power as the filament thins out and its resistance increases. So, this statement is true.

Key concepts

Black-body Radiation

Imagine a scenario where you're cozily tucked under blankets on a chilly night, and you switch on a heater. The glowing coils are not just providing warmth; they're demonstrating an essential phenomenon known as black-body radiation. This is much like the incandescent bulb's filament in our exercise.

Black-body radiation can be described as the thermal electromagnetic radiation within or surrounding a body in thermodynamic equilibrium with its environment, or emitted by a black body held at constant, uniform temperature. A black body itself is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence.

In reality, no material is a perfect black body, but some, like the tungsten filament in our incandescent bulb, come quite close. When heated up, it emits a spectrum of light that ranges from infrared to visible light and even ultraviolet at extremely high temperatures. This radiation is emitted due to the acceleration of charged particles within the filament, which is a consequence of the heat energy overcoming the binding forces within the atoms. An essential characteristic of black-body radiation is its spectrum and intensity are determined only by the temperature of the body, not on its shape or composition.

Resistance and Temperature Relationship

When curling your hair with a flat iron, have you noticed how it becomes increasingly more difficult to run the iron through your hair as it heats up? There's a fascinating similarity here with how resistance changes in a thin tungsten filament as it heats up.

The resistance and temperature relationship in electrical conductors is a critical concept in Physics, particularly in the context of JEE Advanced. It's well known that resistance increases with temperature for most metallic conductors. This is due to the fact that as the temperature rises, the metal's atoms vibrate more vigorously, making it harder for electrons to pass through—the greater the vibrations, the higher the resistance.

This relationship is more evident in our exercise where non-uniform evaporation causes thinning of the filament, contributing to the increase in resistance over time. As such, the resistance over small sections of the filament indeed increases as the filament gets thinner. This is why, in the case of the tungsten filament in a bulb, as it heats up and its constituent atoms vibrate more intensively, the resistance goes up. Further compounding this effect is the non-uniform evaporation, which leads to an uneven distribution of resistance and temperature along the filament's length.

Planck's Law

What happens when you inch your hands closer to a campfire's flames to catch the warmth? What you're soaking in is not just the heat but also the light, and the nature of this light can tell us a lot about the temperature of the fire. Planck’s law gives us a mathematical description of the radiation emitted by a black body as a function of frequency or wavelength, which is essentially in line with what you're experiencing at that campfire.

Planck’s law states that every object emits radiation at a frequency that is dependent on its temperature. The law is mathematically represented with a formula that combines ideas from classical electromagnetic theory with quantum mechanics. It shows that the intensity of radiation increases and the peak of the emitted spectrum shifts to higher frequencies as the temperature of a black body increases.

In the filament of an incandescent bulb, just before it breaks up due to the non-uniform evaporation of tungsten, as per Planck's law, the parts of the filament that are hotter would emit light at higher frequencies. This understanding aligns with statement (C) in our exercise problem, indicating the emission of more light at higher frequencies before the filament breaks up. Planck's law clarifies why, as the temperature of the thinning tungsten filament increases, we see an increase in the frequency of light it emits.