A metal surface of threshold wavelength 550 nm is irradiated with blue light of wavelength 450 nm. Calculate the (1) maximum kinetic energy of the emitted photoelectrons, (ii) reverse voltage required to stop the emitted photoelectrons from reaching the anode. [4] (c) State, with a reason, the effect on the quantity calculated in b(ii) of doubling the (1) intensity of incident light,, (ii) wavelength.
(i) To calculate the maximum kinetic energy of the emitted photoelectrons, we can use the equation:
\(\displaystyle E_{\text{{max}}} = h \times \left( \frac{{c}}{{\lambda_{\text{{threshold}}}}} - \frac{{c}}{{\lambda}} \right) \)
Where:
- \( E_{\text{{max}}} \) is the maximum kinetic energy of the emitted photoelectrons,
- \( h \) is the Planck's constant (\( 6.62607015 \times 10^{-34} \, \text{{J s}} \)),
- \( c \) is the speed of light (\( 3.0 \times 10^8 \, \text{{m/s}} \)),
- \( \lambda_{\text{{threshold}}} \) is the threshold wavelength of the metal surface (\( 550 \times 10^{-9} \, \text{{m}} \)), and
- \( \lambda \) is the wavelength of the incident light (\( 450 \times 10^{-9} \, \text{{m}} \)).
Let's substitute the values in:
\(\displaystyle E_{\text{{max}}} = (6.62607015 \times 10^{-34}) \times \left( \frac{{3.0 \times 10^8}}{{550 \times 10^{-9}}} - \frac{{3.0 \times 10^8}}{{450 \times 10^{-9}}} \right) \)
Solving this equation gives the maximum kinetic energy of the emitted photoelectrons.
(ii) To calculate the reverse voltage required to stop the emitted photoelectrons from reaching the anode, we can use the equation:
\(\displaystyle V = \frac{{E_{\text{{max}}}}}{{q}} \)
Where:
- \( V \) is the reverse voltage,
- \( E_{\text{{max}}} \) is the maximum kinetic energy of the emitted photoelectrons (calculated in (i)),
- \( q \) is the charge of an electron (\( 1.6 \times 10^{-19} \, \text{{C}} \)).
Substituting the values, we can calculate the reverse voltage.
(c)(i) Doubling the intensity of incident light will not affect the maximum kinetic energy of the emitted photoelectrons. According to the equation, the maximum kinetic energy is only dependent on the difference between the threshold wavelength and the incident light wavelength.
(ii) Doubling the wavelength of the incident light will decrease the maximum kinetic energy of the emitted photoelectrons. As seen in the equation, the maximum kinetic energy is inversely proportional to the wavelength of the incident light. Hence, doubling the wavelength will reduce the maximum kinetic energy.
To calculate the maximum kinetic energy of the emitted photoelectrons, we can use the equation:
maximum kinetic energy (KEmax) = Planck's constant (h) x (c / wavelength of incident light - wavelength of threshold)
Given:
Threshold wavelength (λthreshold) = 550 nm
Wavelength of incident blue light (λincident) = 450 nm
Note that Planck's constant (h) = 6.63 x 10^-34 J·s
The speed of light (c) = 3.0 x 10^8 m/s
First, let's convert the given wavelengths from nanometers (nm) to meters (m):
λthreshold = 550 nm = 550 x 10^-9 m
λincident = 450 nm = 450 x 10^-9 m
Now we can calculate the maximum kinetic energy (KEmax):
KEmax = (6.63 x 10^-34 J·s) x (3.0 x 10^8 m/s) / (450 x 10^-9 m - 550 x 10^-9 m)
KEmax = (6.63 x 10^-34 J·s) x (3.0 x 10^8 m/s) / (100 x 10^-9 m)
KEmax = (6.63 x 10^-34 J·s) x (3.0 x 10^8 m/s) / (10^-7 m)
KEmax = (6.63 x 3.0) x 10^-34+8-7 J
KEmax ≈ 19.89 x 10^-26 J
Therefore, the maximum kinetic energy of the emitted photoelectrons is approximately 19.89 x 10^-26 J.
Next, to calculate the reverse voltage required to stop the emitted photoelectrons from reaching the anode, we can use the equation:
reverse voltage (Vreverse) = KEmax / elementary charge (e)
Note that the elementary charge (e) = 1.6 x 10^-19 C (Coulombs)
Vreverse = (19.89 x 10^-26 J) / (1.6 x 10^-19 C)
Vreverse ≈ 12.43 x 10^-7 V
Therefore, the reverse voltage required to stop the emitted photoelectrons from reaching the anode is approximately 12.43 x 10^-7 V.
Now let's consider the effect of doubling the (1) intensity of incident light and (ii) wavelength on the reverse voltage calculated in b(ii):
(i) Doubling the intensity of incident light:
Doubling the intensity of incident light would increase the number of photons incident on the metal surface, resulting in more photoelectrons being emitted. However, it will not have any effect on the maximum kinetic energy or the reverse voltage required to stop the emitted photoelectrons.
(ii) Doubling the wavelength:
Doubling the wavelength of the incident light means that the new wavelength would be 2 times the original value (2λincident). This change in wavelength would directly affect the maximum kinetic energy and the reverse voltage required to stop the emitted photoelectrons.
To recalculate the maximum kinetic energy (KEmax) and the reverse voltage (Vreverse), we need to substitute the new wavelength value (2λincident) into the original formulas.