© All Rights reserved @ LearnWithDash
Step-by-Step Solution
Step 1: Identify the Relevant Physical Principles
The question pertains to the photoelectric effect, where electrons are emitted when light of sufficient energy strikes a metal surface. The key formula for the maximum kinetic energy of the emitted electrons is:
$K_{\max} = \frac{hc}{\lambda_{\text{incident}}} - \frac{hc}{\lambda_{\text{threshold}}}$
Alternatively, in electronvolts (eV), using the relation $E(\text{in eV}) = \frac{1237}{\lambda (\text{in nm})}$, we can write:
$K_{\max} = \left[\frac{1237}{\lambda_{\text{incident}}}\right] - \left[\frac{1237}{\lambda_{\text{threshold}}}\right]$
Step 2: Substitute the Given Wavelength Values
We are given:
Threshold wavelength, $\lambda_{\text{threshold}} = 380 \text{ nm}$
Incident wavelength, $\lambda_{\text{incident}} = 260 \text{ nm}$
Thus,
$K_{\max} = 1237 \left(\frac{1}{260} - \frac{1}{380}\right)\,\text{eV}$
Step 3: Simplify the Expression
Combine the terms inside the parentheses:
$\frac{1}{260} - \frac{1}{380} = \frac{380 - 260}{260 \times 380} = \frac{120}{260 \times 380}
$
Therefore,
$
K_{\max} = 1237 \times \frac{120}{260 \times 380}\,\text{eV}
$
Step 4: Perform the Calculation
Calculate the numerator and denominator:
$260 \times 380 = 98800
$
So,
$
K_{\max} = 1237 \times \frac{120}{98800}\,\text{eV}
$
Simplifying this will yield:
$
K_{\max} \approx 1.5\,\text{eV}
$
Step 5: State the Final Answer
The maximum kinetic energy of the emitted photoelectrons is therefore 1.5 eV.
