Question

The filament temperature of a light bulb is 2 000 K when the bulb delivers 40 W of power.

If its emissivity remains constant, what power is delivered when the filament temperature is 2 500 K?

a. 105 W

b. 62 W

c. 98 W

d. 50 W

Answer 1

Using the Stefan-Boltzmann law, we calculate the power delivered by the light bulb when the filament temperature rises from 2000 K to 2500 K, while the emissivity remains constant. The new power output is approximately 98 W, which is the answer (c).

Step-by-step explanation:

The subject of this question is Physics, and it relates to the concept of blackbody radiation and the power emitted by a heated object according to the Stefan-Boltzmann law. The Stefan-Boltzmann law states that the power radiated per unit area of a blackbody is directly proportional to the fourth power of the blackbody's temperature, given by the equation P = εσT^4, where ε is the emissivity of the material, σ is the Stefan-Boltzmann constant, and T is the temperature in kelvins.

To solve the problem provided in the question, we'll assume that the emissivity of the filament remains constant. We can set up a ratio of the powers based on the Stefan-Boltzmann law:

Power1/T1^4 = Power2/T2^4

Solving for Power2 when the original power (Power1) is 40 W and the original temperature (T1) is 2000 K and the new temperature (T2) is 2500 K:

Power2 = Power1 × (T2/T1)^4

Power2 = 40 W × (2500/2000)^4

Power2 = 40 W × (5/4)^4

Power2 = 40 W × (625/256)

Power2 = 40 W × 2.44140625

Power2 = 97.65625 W

So, the power delivered when the filament temperature is 2500 K is approximately 98 W, which corresponds to option (c).

Answer 2

c. 98 W

Step-by-step explanation:

`P_(i)` = initial power delivered by bulb at initial temperature = 40 W

`T_(i)` = initial temperature of filament = 2000 K

Now the temperature of filament of the bulb rises

`P_(f)` =Final power delivered by bulb at higher temperature

`T_(f)` = Final higher temperature of filament = 2500 K

As per Stefan's law, the power delivered by the bulb depends directly on fourth power of the temperature. hence we can write

`(P_(f))/(P_(i)) = (T_(f)^(4) )/(T_(i)^(4)) \\(P_(f))/(40) = (2500^(4) )/(2000^(4))\\\\P_(f) = (2.44) (40)\\P_(f) = 98 W`