Question

A 50 watt bulb emits monochromatic red light of wavelength of 795 nm. The number of photons emitted per second by the bulb is x × 10$^{-20}$. The value of x is______. [Given: h = 6.63 × 10$^{-34}$ Js and c = 3.0 × 10$^8$ ms$^{-1}$] [1 Sept, 2021 (Shift-II)]

Answer 1

2

Answer 2

The power of the bulb is given as $P = 50$ watt, which means it emits 50 Joules of energy per second.
The light emitted is monochromatic red light with a wavelength $\lambda = 795$ nm.
We are given Planck's constant $h = 6.63 \times 10^{-34}$ Js and the speed of light $c = 3.0 \times 10^8$ ms$^{-1}$.

The energy of a single photon is given by the formula: $E = \frac{hc}{\lambda}$

First, convert the wavelength from nanometers to meters: $\lambda = 795 \text{ nm} = 795 \times 10^{-9} \text{ m}$

Now, calculate the energy of one photon: $E = \frac{(6.63 \times 10^{-34} \text{ Js}) \times (3.0 \times 10^8 \text{ ms}^{-1})}{795 \times 10^{-9} \text{ m}}$ $E = \frac{6.63 \times 3.0}{795} \times 10^{-34 + 8 - (-9)} \text{ J}$ $E = \frac{19.89}{795} \times 10^{-34 + 8 + 9} \text{ J}$ $E = \frac{19.89}{795} \times 10^{-17} \text{ J}$

The number of photons emitted per second, $N$, is equal to the total energy emitted per second (power $P$) divided by the energy of a single photon ($E$): $N = \frac{P}{E}$ $N = \frac{50 \text{ J/s}}{\frac{19.89}{795} \times 10^{-17} \text{ J}}$ $N = \frac{50 \times 795}{19.89} \times 10^{17} \text{ s}^{-1}$ $N = \frac{39750}{19.89} \times 10^{17} \text{ s}^{-1}$

Now, perform the division: $\frac{39750}{19.89} \approx 1998.4917$

So, $N \approx 1998.4917 \times 10^{17} \text{ s}^{-1}$

The question states that the number of photons emitted per second is $x \times 10^{-20}$. Based on typical calculations and the similar question provided, the exponent 10$^{-20}$ is highly likely a typo and should be 10$^{20}$. Assuming the question meant $x \times 10^{20}$:

We have $N \approx 1998.4917 \times 10^{17}$. We need to express this in the form $x \times 10^{20}$. $N = 1998.4917 \times 10^{17} = 1998.4917 \times 10^{17} \times \frac{10^3}{10^3} = \frac{1998.4917}{10^3} \times 10^{17} \times 10^3$ $N = 1.9984917 \times 10^{20}$

So, $1.9984917 \times 10^{20} = x \times 10^{20}$. This gives $x = 1.9984917$.

Since this is likely an integer answer type question, we should round $x$ to the nearest integer. $x \approx 1.9984917$ rounded to the nearest integer is 2.

If we strictly follow the given exponent $10^{-20}$, then: $1.9984917 \times 10^{20} = x \times 10^{-20}$ $x = \frac{1.9984917 \times 10^{20}}{10^{-20}} = 1.9984917 \times 10^{20 - (-20)} = 1.9984917 \times 10^{40}$ This value of $x$ is extremely large and unrealistic in this context. Therefore, we proceed with the assumption that the exponent was intended to be 20.

Assuming the question intended $x \times 10^{20}$, the value of $x$ is approximately 1.9984917. Rounding to the nearest integer gives $x=2$.

The final answer is 2.

Explanation of the solution:

1. Calculate the energy of a single photon using the formula $E = \frac{hc}{\lambda}$, converting wavelength to meters.
2. Calculate the number of photons emitted per second by dividing the total power of the bulb by the energy of a single photon: $N = \frac{P}{E}$.
3. Express the calculated number of photons $N$ in the form $x \times 10^{20}$ (assuming the likely typo in the question's exponent) to find the value of $x$.
4. Round the value of $x$ to the nearest integer as expected for an integer answer type question.