Question

The work functions of Silver and Sodium are \(4.6\) and \(2.3 \mathrm{eV}\), respectively. The ratio of the slope of the stopping potential versus frequency plot for Silver to that of Sodium is ____ .

Short answer

The ratio of the slope of the stopping potential versus frequency plot for Silver to that of Sodium is 1.

Step-by-step solution

Understand the Photoelectric Effect Equation

The photoelectric effect can be described by the equation: \(KE_{max} = hu - \phi\), where \(KE_{max}\) is the maximum kinetic energy of the ejected electrons, \(h\) is Planck's constant, \(u\) is the frequency of the incident light, and \(\phi\) is the work function of the material. The stopping potential \(V_s\) is related to the \(KE_{max}\) by \(eV_s = KE_{max}\), where \(e\) is the charge of an electron. Combining these two relations gives \(eV_s = hu - \phi\).

Derive the Slope of the Stopping Potential versus Frequency Plot

To find the slope of the stopping potential versus frequency plot, we need to consider the linear equation \(eV_s = hu - \phi\). The slope \(m\) of this equation when graphed is given by \(m = \frac{\Delta eV_s}{\Delta u} = h\), where \(h\) is a constant (Planck's constant) for all materials.

Calculate the Ratio of the Slopes for Silver and Sodium

Given that the slope is the same for any material, since it's equal to Planck's constant, the ratio of the slopes for Silver and Sodium is simply \(\frac{m_{Ag}}{m_{Na}} = \frac{h}{h} = 1\), where \(m_{Ag}\) and \(m_{Na}\) are the slopes for Silver and Sodium, respectively.

Key concepts

Stopping Potential in the Photoelectric Effect

The concept of stopping potential is intrinsic to the photoelectric effect, a phenomenon that entails the emission of electrons from a material when it is exposed to light. This process is not merely about the liberation of electrons but involves specific energies that enable their ejection. The stopping potential, denoted as \( V_s \), is the minimum voltage necessary to halt the fastest-moving photoelectron emitted from a material.

Such a potential is crucial because it helps us measure the maximum kinetic energy, \( KE_{max} \), of the ejected electrons since \( eV_s = KE_{max} \). When light of a certain frequency strikes a material with a known work function, it's the stopping potential that allows us to calculate the energy input from the light as well. In essence, the stopping potential creates a balance; it prevents any photoelectron, regardless of its kinetic energy, from reaching the anode within a phototube, resulting in a current of zero. This key principle is fundamental to understanding how light interacts with matter on a quantum level.

Work Function & Its Role in the Photoelectric Effect

In the maze of electrons and energies, the work function is a pivotal factor that governs the photoelectric effect. Often symbolized as \( \$ \), the work function is defined as the minimum energy required to liberate an electron from the surface of a metal or any material. This varies from metal to metal; for instance, the work functions of Silver and Sodium are \(4.6\) eV and \(2.3\) eV, respectively. The work function is intrinsic to each material and relates to how tightly an electron is bound to the material's surface.

This concept directly ties into the basic photoelectric equation, \( KE_{max} = hu - \$ \), where \( KE_{max} \) represents the maximum kinetic energy of the ejected electrons, and \( hu \) signifies the energy of the photons striking the surface. With different materials showing varying resistances to electron ejection, the work function is pivotal in determining not just if the photoelectric effect will occur but how efficiently it will proceed once the incoming photon energy exceeds this threshold value.

Decoding Planck's Constant

Dive into the world of quantum mechanics, and you will find Planck's constant, symbolized as \( h \), playing the role of one of the fundamental building blocks. This constant is a bedrock in the realm of physics, as it relates the energy of a photon to the frequency of the light. With an established value of approximately \(6.626 \times 10^{-34} \) Joule seconds, Planck's constant bridges the microscopic and macroscopic realms, making sense of the quantization of energy in the universe.

In our photoelectric context, Planck's constant is reflected in the slope when we graph the stopping potential against the frequency of incident light: the linear relationship is represented as \( eV_s = hu - \$ \). In this equation, the slope of this line elucidates Planck's constant directly, which is astounding as it means that no matter the material, the slope pertaining to the stopping potential versus frequency plot is constant and equal to \( h \), underlining the universal applicability of this fundamental constant.