Answer :
To find the speed of an electron dislodged from lead using ultraviolet light, we need to use a few principles from Einstein's photoelectric theory.
Here's how you can solve the problem step-by-step:
1. Identify the Given Values:
- Threshold energy for lead, [tex]\( \phi = 6.81 \times 10^{-19} \)[/tex] J.
- Wavelength of ultraviolet light, [tex]\( \lambda = 51.7 \)[/tex] nm [tex]\( = 51.7 \times 10^{-9} \)[/tex] m.
- Speed of light, [tex]\( c = 3.00 \times 10^8 \)[/tex] m/s.
- Planck’s constant, [tex]\( h = 6.626 \times 10^{-34} \)[/tex] J·s.
- Mass of an electron, [tex]\( m = 9.109 \times 10^{-31} \)[/tex] kg.
2. Calculate the Energy of a Photon:
The energy of a photon can be calculated using the formula:
[tex]\[ E = \frac{hc}{\lambda} \][/tex]
Plug in the values:
[tex]\[
E = \frac{6.626 \times 10^{-34} \times 3.00 \times 10^8}{51.7 \times 10^{-9}} \approx 3.845 \times 10^{-18} \text{ J}
\][/tex]
3. Calculate the Kinetic Energy of the Electron:
According to the photoelectric equation, the kinetic energy (K.E.) of the dislodged electron is the difference between the photon's energy and the threshold energy:
[tex]\[ \text{K.E.} = E - \phi \][/tex]
[tex]\[
\text{K.E.} = 3.845 \times 10^{-18} - 6.81 \times 10^{-19} \approx 3.164 \times 10^{-18} \text{ J}
\][/tex]
4. Calculate the Speed of the Electron:
Use the kinetic energy formula for the electron to find its speed:
[tex]\[ \text{K.E.} = \frac{1}{2} m v^2 \][/tex]
Solve for [tex]\( v \)[/tex]:
[tex]\[
v = \sqrt{\frac{2 \cdot \text{K.E.}}{m}}
\][/tex]
[tex]\[
v = \sqrt{\frac{2 \times 3.164 \times 10^{-18}}{9.109 \times 10^{-31}}} \approx 2,635,659 \text{ m/s}
\][/tex]
Therefore, the speed of the dislodged electron is approximately 2,635,659 meters per second.
Here's how you can solve the problem step-by-step:
1. Identify the Given Values:
- Threshold energy for lead, [tex]\( \phi = 6.81 \times 10^{-19} \)[/tex] J.
- Wavelength of ultraviolet light, [tex]\( \lambda = 51.7 \)[/tex] nm [tex]\( = 51.7 \times 10^{-9} \)[/tex] m.
- Speed of light, [tex]\( c = 3.00 \times 10^8 \)[/tex] m/s.
- Planck’s constant, [tex]\( h = 6.626 \times 10^{-34} \)[/tex] J·s.
- Mass of an electron, [tex]\( m = 9.109 \times 10^{-31} \)[/tex] kg.
2. Calculate the Energy of a Photon:
The energy of a photon can be calculated using the formula:
[tex]\[ E = \frac{hc}{\lambda} \][/tex]
Plug in the values:
[tex]\[
E = \frac{6.626 \times 10^{-34} \times 3.00 \times 10^8}{51.7 \times 10^{-9}} \approx 3.845 \times 10^{-18} \text{ J}
\][/tex]
3. Calculate the Kinetic Energy of the Electron:
According to the photoelectric equation, the kinetic energy (K.E.) of the dislodged electron is the difference between the photon's energy and the threshold energy:
[tex]\[ \text{K.E.} = E - \phi \][/tex]
[tex]\[
\text{K.E.} = 3.845 \times 10^{-18} - 6.81 \times 10^{-19} \approx 3.164 \times 10^{-18} \text{ J}
\][/tex]
4. Calculate the Speed of the Electron:
Use the kinetic energy formula for the electron to find its speed:
[tex]\[ \text{K.E.} = \frac{1}{2} m v^2 \][/tex]
Solve for [tex]\( v \)[/tex]:
[tex]\[
v = \sqrt{\frac{2 \cdot \text{K.E.}}{m}}
\][/tex]
[tex]\[
v = \sqrt{\frac{2 \times 3.164 \times 10^{-18}}{9.109 \times 10^{-31}}} \approx 2,635,659 \text{ m/s}
\][/tex]
Therefore, the speed of the dislodged electron is approximately 2,635,659 meters per second.
