Threshold Frequency and Wavelength Calculation:
- Threshold Frequency Calculation:
The threshold frequency is the minimum frequency of light required to eject an electron from a metal surface. It can be calculated using the formula:
\( f_{\text{threshold}} = \dfrac{W}{h} \), where:
\( f_{\text{threshold}} \) = threshold frequency,
\( W \) = work function of the metal (4 x 10^-19 J),
\( h \) = Planck's constant (6.63 x 10^-34 J.s).
Substitute the given values into the formula:
\( f_{\text{threshold}} = \dfrac{4 \times 10^{-19}}{6.63 \times 10^{-34}} \)
Calculating the threshold frequency will give you the value in Hz.
- Threshold Wavelength Calculation:
The threshold wavelength can be calculated using the formula:
\( \lambda = \dfrac{c}{f_{\text{threshold}}} \), where:
\( \lambda \) = threshold wavelength,
\( c \) = speed of light in vacuum (3 x 10^8 m/s),
\( f_{\text{threshold}} \) = threshold frequency (calculated in the previous step).
Substitute the known values into the formula:
\( \lambda = \dfrac{3 \times 10^8}{f_{\text{threshold}}} \)
Calculating the threshold wavelength will give you the value in meters.

Explanation:
The work function of a metal is the minimum amount of energy required to remove an electron from its surface. The threshold frequency is directly proportional to the work function and inversely proportional to the threshold wavelength. By calculating these values, we can determine the characteristics of the metal in terms of its interaction with light.