Question

Light of \(300 \mathrm{nm}\) wavelength strikes a metal plate, and photoclectrons are produced moving as \(f\) ast as \(0.002 c\). (a) What is the work function of the metal? (b) What is the threshold wavelength for this metal?

Short answer

The work function of the metal is \(4.99 × 10^{-19} J\). The threshold wavelength for this metal is \(398.5 nm\).

Step-by-step solution

Calculate the Energy of the incident light

The formula to calculate energy (E) from the wavelength of light is given by: \(E=h \cdot c / \lambda\) where, \(h\) is Planck's constant, \(c\) is the speed of light, and \( \lambda \) is the wavelength of light. Substituting the known values \(h = 6.63 × 10^{-34} Js\), \(c = 3 × 10^8 m/s\), and the known wavelength \( \lambda = 300nm = 300 × 10^{-9} m\), the energy value can therefore be computed as follows: \(E = (6.63 × 10^{-34} Js) \cdot (3 ×10^8 m/s) / (300 × 10^{-9} m) = 6.63 × 10^{-19} J.\)

Find the kinetic energy of ejected electrons

The kinetic energy of the photoelectrons is calculated by using the formula: \(KE = 1/2mv^2\), where \(m\) is the electron mass and \(v\) is their velocity. Since the velocity is given in terms of speed of light \(c\) as \(0.002c\), we can substitute \(m = 9.11 × 10^{-31} kg\) and \(v = 0.002 × 3 × 10^8 m/s\). Thus, \(KE = 1/2 \cdot 9.11×10^{-31}kg \cdot (0.002 × 3 × 10^8 m/s)^2 = 1.64 × 10^{-19} J.\)

Calculate the work function

The work function (W) can be obtained from the difference between the incident energy and the kinetic energy of the emitted electrons. So, \(W = E - KE\). Substituting the values from steps 1 and 2 respectively, \(W = 6.63 × 10^{-19} J - 1.64 × 10^{-19} J = 4.99 × 10^{-19} J.\)

Calculate the threshold wavelength

The threshold wavelength (\(\lambda_t\)) is the maximum wavelength at which photoelectric effect occurs. It can be calculated using the formula: \(\lambda_t = h \cdot c / W\), where \(W\) is the work function obtained in step 3. Substituting the values, \(\lambda_t = (6.63 × 10^{-34} Js \cdot 3 × 10^8 m/s) / (4.99 × 10^{-19} J) = 398.5 nm.\)

Key concepts

Work Function

In the context of the photoelectric effect, the work function is a crucial concept. It is essentially the minimum energy required to release electrons from a metal surface. This energy barrier must be overcome by the incoming light's photons for the electrons to escape.

The work function is different for each material and is a constant specific to that particular metal. In the mathematical realm, it's represented by the symbol \( W \) and is measured in Joules (J) or electronvolts (eV).

• The work function provides insight into how a metal interacts with light.
• For photoelectric emission to occur, the energy of incoming photons \( E \) must be greater than the work function \( W \).
• If the photon's energy is exactly equal to the work function, the emitted electron will barely escape from the surface.

The work function is determined using the equation: \[ W = E - KE \]where \( E \) is the energy of the incident photons, and \( KE \) is the kinetic energy of the emitted electrons. This equation highlights that not all the photon's energy translates into kinetic energy, as it must first overcome the work function barrier.

Threshold Wavelength

The threshold wavelength relates directly to the work function. It is the maximum wavelength of light that is capable of ejecting electrons from a metal surface and can still cause the photoelectric effect.

At this specific wavelength, the energy of the incoming photons is equal to the work function. This means the electrons have just enough energy to escape the metal's surface but no excess kinetic energy to speed them up.

• Beyond the threshold wavelength, the energy of photons is insufficient to overcome the work function.
• Materials with a lower work function typically have a longer threshold wavelength.
• The threshold wavelength is essential for determining the color or type of light needed to initiate the photoelectric effect in a material.

The formula to find this wavelength is given by:\[ \lambda_t = \frac{h \cdot c}{W} \]where \( h \) is Planck's constant, \( c \) is the speed of light, and \( W \) is the work function. This formula highlights that a larger work function corresponds to a shorter threshold wavelength.

Kinetic Energy

Kinetic energy in the context of the photoelectric effect describes the energy possessed by the emitted electrons as they move after being ejected.

Once a photoelectron escapes the attraction of a metal surface, the remainder of the photon's energy (after subtracting the work function) manifests as the kinetic energy of the electron. It's a measure of how fast the electrons are traveling after liberation.

• Kinetic energy is influenced by the energy of the incoming photons and the material's work function.
• If the incident light's energy is just above the work function, the emitted electrons will have low kinetic energy.
• The formula to compute kinetic energy is \( KE = \frac{1}{2}mv^2 \).

In scenarios involving the speed of light, the photoelectron's velocity \( v \) is often given as a fraction of the speed of light (\( c \)). Substituting into the kinetic energy equation, this allows us to calculate the energy that electrons have after being freed from the metal.