To solve for the minimum frequency of the radiation required to emit electrons from a metallic surface given the maximum kinetic energy, we use the photoelectric effect equation:
[tex]\[ KE_{\text{max}} = h \cdot f - h \cdot f_{\text{min}} \][/tex]
where:
- [tex]\( KE_{\text{max}} \)[/tex] is the maximum kinetic energy of the emitted electrons.
- [tex]\( h \)[/tex] is Planck's constant.
- [tex]\( f \)[/tex] is the frequency of the incoming radiation.
- [tex]\( f_{\text{min}} \)[/tex] is the minimum frequency needed to emit electrons.
Given data:
- [tex]\( KE_{\text{max}} = 1.6 \times 10^{-19} \, \text{J} \)[/tex]
- [tex]\( f = 7.5 \times 10^{14} \, \text{Hz} \)[/tex]
- [tex]\( h = 6.6 \times 10^{-34} \, \text{Js} \)[/tex]
First, using the photoelectric equation:
[tex]\[ 1.6 \times 10^{-19} \, \text{J} = (6.6 \times 10^{-34} \, \text{Js}) \cdot (7.5 \times 10^{14} \, \text{Hz}) - (6.6 \times 10^{-34} \, \text{Js}) \cdot f_{\text{min}} \][/tex]
We substitute the given values into the equation:
[tex]\[ 1.6 \times 10^{-19} = (6.6 \times 10^{-34}) \cdot (7.5 \times 10^{14}) - (6.6 \times 10^{-34}) \cdot f_{\text{min}} \][/tex]
To isolate [tex]\( f_{\text{min}} \)[/tex], arrange the equation as follows:
[tex]\[ 1.6 \times 10^{-19} = (6.6 \times 10^{-34} \cdot 7.5 \times 10^{14}) - (6.6 \times 10^{-34} \cdot f_{\text{min}}) \][/tex]
Calculate [tex]\( 6.6 \times 10^{-34} \cdot 7.5 \times 10^{14} \)[/tex]:
[tex]\[ 6.6 \times 10^{-34} \cdot 7.5 \times 10^{14} = 4.95 \times 10^{-19} \][/tex]
So the equation now is:
[tex]\[ 1.6 \times 10^{-19} = 4.95 \times 10^{-19} - 6.6 \times 10^{-34} \cdot f_{\text{min}} \][/tex]
Rearranging it to solve for [tex]\( f_{\text{min}} \)[/tex]:
[tex]\[ 6.6 \times 10^{-34} \cdot f_{\text{min}} = 4.95 \times 10^{-19} - 1.6 \times 10^{-19} \][/tex]
[tex]\[ 6.6 \times 10^{-34} \cdot f_{\text{min}} = 3.35 \times 10^{-19} \][/tex]
[tex]\[ f_{\text{min}} = \frac{3.35 \times 10^{-19}}{6.6 \times 10^{-34}} \][/tex]
[tex]\[ f_{\text{min}} = 5.075757575757576 \times 10^{14} \, \text{Hz} \][/tex]
Therefore, the minimum frequency of the radiation required to emit electrons from the metallic surface is approximately:
[tex]\[ f_{\text{min}} \approx 5.08 \times 10^{14} \, \text{Hz} \][/tex]
