Suppose the `10^19` photons emitted per second from the 100 W lightbulb in were all focused onto a piece of black paper and absorbed. (a)Calculate the momentum of one photon and (b)estimate the force all these photons could exert on the paper.

Explanation

To solve the problem step by step, we will break it down into two parts as per the questions asked.

### Part (a): Calculate the momentum of one photon.

1. **Given Data**:
- Power of the light bulb, \( P = 100 \, \text{W} \)
- Number of photons emitted per second, \( n = 10^{19} \)

2. **Calculate the total energy emitted per second**:
\[ \text{Energy} (E) = \text{Power} \times \text{Time} = 100 \, \text{W} \times 1 \, \text{s} = 100 \, \text{J} \]

3. **Relate energy to the number of photons**:
The energy of one photon can be expressed as:
\[ E = n \cdot E_{\text{photon}} \implies E_{\text{photon}} = \frac{E}{n} = \frac{100 \, \text{J}}{10^{19}} = 10^{-18} \, \text{J} \]

4. **Use the energy of a photon to find its wavelength**:
The energy of a photon is also given by:
\[ E_{\text{photon}} = \frac{hc}{\lambda} \]
Rearranging gives:
\[ \lambda = \frac{hc}{E_{\text{photon}}} \]
Where \( h \) (Planck's constant) is approximately \( 6.626 \times 10^{-34} \, \text{Js} \) and \( c \) (speed of light) is approximately \( 3 \times 10^8 \, \text{m/s} \).

5. **Calculate the wavelength**:
\[ \lambda = \frac{(6.626 \times 10^{-34} \, \text{Js}) \times (3 \times 10^8 \, \text{m/s})}{10^{-18} \, \text{J}} = 1.9878 \times 10^{-7} \, \text{m} \approx 2 \times 10^{-7} \, \text{m} \]

6. **Calculate the momentum of one photon**:
The momentum \( p \) of a photon is given by:
\[ p = \frac{h}{\lambda} \]
Substituting the values:
\[ p = \frac{6.626 \times 10^{-34} \, \text{Js}}{2 \times 10^{-7} \, \text{m}} \approx 3.313 \times 10^{-27} \, \text{kg m/s} \]

### Part (b): Estimate the force all these photons could exert on the paper.

1. **Calculate the total momentum of all photons emitted per second**:
The total momentum \( P_{\text{total}} \) from all photons is:
\[ P_{\text{total}} = n \cdot p = 10^{19} \cdot 3.313 \times 10^{-27} \approx 3.313 \times 10^{-8} \, \text{kg m/s} \]

2. **Calculate the force exerted by the photons**:
The force \( F \) is given by the rate of change of momentum:
\[ F = \frac{\Delta P}{\Delta t} = \frac{P_{\text{total}}}{1 \, \text{s}} = 3.313 \times 10^{-8} \, \text{N} \]

### Final Answers:
- (a) The momentum of one photon is approximately \( 3.313 \times 10^{-27} \, \text{kg m/s} \).
- (b) The force exerted by all these photons on the paper is approximately \( 3.313 \times 10^{-8} \, \text{N} \).