## **21.2 PHOTOELECTRIC EFFECT** The photoelectric effect is the phenomenon where electrons are emitted from a metal surface when it is exposed to electromagnetic radiation of a sufficiently high frequency. The term "photoelectric" combines "photo" (light) and "electric" (electron), signifying the interaction between photons and electrons. The emitted electrons are called **photoelectrons**. --- ### **Experimental Setup** An evacuated glass tube contains two metal plates. One plate (cathode) is connected to the negative terminal of a battery, and the other (anode/detector) is connected to the positive terminal. - In the dark, no current flows. - When light of an appropriate frequency illuminates the cathode, the ammeter detects a current, indicating that electrons are emitted from the cathode and collected by the anode. ### **Factors Affecting Photoelectric Effect** 1. **Effect of Applied Voltage:** - For a given light intensity, as the positive potential difference (V) increases, the photoelectric current increases until it reaches a saturation point, where all emitted photoelectrons are collected. - If the potential is reversed (V is negative), the current decreases. - **Stopping Potential ($V_0$)**: The specific negative potential at which the photoelectric current becomes zero. At this potential, even the most energetic electrons are repelled and cannot reach the anode. - The maximum kinetic energy ($K.E_{max}$) of the photoelectrons is related to the stopping potential by the equation: $$ K.E_{max} = eV_0 $$ 2. **Effect of Light Intensity:** - The photoelectric current is directly proportional to the intensity of the incident light (for a constant frequency). - This implies that the **number of photoelectrons emitted per second is proportional to the intensity** of the incident radiation. - However, the maximum kinetic energy of the photoelectrons is *independent* of the light intensity. This contradicts classical wave theory, which predicts that more intense light (higher amplitude) should impart more energy to the electrons. 3. **Effect of Light Frequency:** - The maximum kinetic energy of photoelectrons is directly proportional to the frequency of the incident radiation. - **Threshold Frequency ($f_0$)**: For any given metal, there is a minimum frequency, called the threshold frequency, below which no photoelectrons are emitted, regardless of the light's intensity. - This is another observation that classical physics cannot explain. ### **Work Function and Einstein's Photon Theory** - **Work Function ($\Phi$):** The minimum amount of energy required for an electron to escape from a metal surface. It is a characteristic property of the metal. - In 1905, Albert Einstein explained the photoelectric effect by proposing that light consists of discrete energy quanta called **photons**. The energy of a single photon is given by $E = hf$. - **Einstein's Photoelectric Equation:** When a photon strikes an electron, it transfers its entire energy. A part of this energy ($\Phi$) is used to overcome the binding forces and escape the metal, while the remainder becomes the electron's maximum kinetic energy ($K.E_{max}$). $$ E = \Phi + K.E_{max} $$ This can also be written as: $$ hf = hf_0 + K.E_{max} $$ or $$ K.E_{max} = hf - \Phi $$ where $\Phi = hf_0$ is the work function. - **Conditions for Photoemission:** - If $E < \Phi$ (or $f < f_0$): No photoelectric effect occurs. - If $E = \Phi$ (or $f = f_0$): Photoelectric effect occurs, but $K.E_{max} = 0$. - If $E > \Phi$ (or $f > f_0$): Photoelectric effect occurs, and the emitted electron has kinetic energy. --- ## **Possible Questions/Answer** - **Q:** A metal whose work function is 4.2 eV is irradiated by radiation whose wavelength is $2000 \AA$. What is the maximum kinetic energy of the emitted electron? **A:** First, calculate the energy of the incident photon: $E = \frac{hc}{\lambda} = \frac{(6.63 \times 10^{-34} \text{ J s})(3 \times 10^{8} \text{ m/s})}{2 \times 10^{-7} \text{ m}} = 9.945 \times 10^{-19} \text{ J}$. The work function is $\Phi = 4.2 \text{ eV} = 6.72 \times 10^{-19} \text{ J}$. Using Einstein's equation, $K.E_{max} = E - \Phi = (9.945 - 6.72) \times 10^{-19} \text{ J} = 3.225 \times 10^{-19} \text{ J}$. Converting to eV: $K.E_{max} = \frac{3.225 \times 10^{-19} \text{ J}}{1.6 \times 10^{-19} \text{ J/eV}} \approx 2.015 \text{ eV}$. - **Q:** Radiation of wavelength $3000 \AA$ falls on a photoelectric surface for which the work function is 1.6 eV. What is the stopping potential for the emitted electron? **A:** Photon energy $E = \frac{hc}{\lambda} = \frac{(6.63 \times 10^{-34})(3 \times 10^8)}{3000 \times 10^{-10}} \approx 6.63 \times 10^{-19} \text{ J} \approx 4.14 \text{ eV}$. $K.E_{max} = E - \Phi = 4.14 \text{ eV} - 1.6 \text{ eV} = 2.54 \text{ eV}$. Since $K.E_{max} = eV_0$, the stopping potential is $V_0 = 2.54 \text{ V}$. | Key Formulas | Description | | :--- | :--- | | $E = \Phi + K.E_{max}$ | Einstein's photoelectric equation, relating incident photon energy (E), work function ($\Phi$), and max kinetic energy. | | $\Phi = hf_0$ | The work function, defined by the metal's threshold frequency ($f_0$). | | $K.E_{max} = eV_0$ | Relates the maximum kinetic energy of a photoelectron to the stopping potential ($V_0$). | | $K.E_{max} = hf - \Phi$ | A rearranged form of Einstein's equation used to calculate the kinetic energy of photoelectrons. | The photoelectric effect was pivotal in establishing the particle nature of light and is the working principle behind devices like photocells, solar panels, and light sensors.