Problem 16 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

A stream of electrons from a heated filaments was passed two charged plates kept at a potential difference \(\mathrm{V}\) esu. If 'e' and \(m\) are charge and mass of an electron, respectively, then the value of \(h / \lambda\) (where \(\lambda\) is wavelength associated with electron wave) is given by: (a) \(\sqrt{m \mathrm{eV}}\) (b) \(\sqrt{2 m \mathrm{eV}}\) (c) \(m \mathrm{eV}\) (d) \(2 \mathrm{meV}\)

Step-by-Step Solution

Answer

The answer is (b) \( \sqrt{2 m \mathrm{eV}} \).

Step 1: Recall the de Broglie wavelength formula

The de Broglie wavelength \( \lambda \) of a particle is given by \( \lambda = \frac{h}{p} \), where \( p \) is the momentum of the particle.

Step 2: Relate kinetic energy to momentum

The kinetic energy of an electron, when accelerated through a potential difference \( V \), is given by \( \, \frac{1}{2}mv^2 = eV \, \). The momentum \( p \) is given by \( p = mv \).

Step 3: Substitute to find momentum

From the equation \( eV = \frac{1}{2}mv^2 \), solve for \( v \) to get \( v = \sqrt{\frac{2eV}{m}} \). Substituting \( v \) into \( p = mv \), we find \( p = m \sqrt{\frac{2eV}{m}} \).

Step 4: Simplify for momentum

Further simplifying \( p = m \sqrt{\frac{2eV}{m}} \), we get \( p = \sqrt{2meV} \), as the mass \( m \) cancels out under the square root.

Step 5: Calculate \( h/\lambda \)

Substitute \( p = \sqrt{2meV} \) into the de Broglie equation \( h/\lambda = p \) to get \( h/\lambda = \sqrt{2meV} \).

Step 6: Conclusion with option selection

The value of \( h/\lambda \) in terms of \( m \), \( e \), and \( V \) is \( \sqrt{2meV} \), which corresponds to option (b).

Key Concepts

Electron Momentum

In the realm of particle physics, momentum is a key concept that describes the motion of particles like electrons. Momentum is a measure of the amount of motion an object has and is the product of its mass and velocity. For electrons, which are subatomic particles, their momentum (\( p \) ) is calculated using the formula:

• Momentum (\( p \) ) = Mass (\( m \) ) × Velocity (\( v \) )

When electrons move with a certain speed, they possess momentum based on their mass and velocity. This is crucial in calculating the de Broglie wavelength, which describes the wave nature of particles like electrons.The momentum is particularly important when electrons gain energy through acceleration, as seen when they pass between charged plates. As the electrons are accelerated, their velocity and thus their momentum changes, influencing their de Broglie wavelength.

Kinetic Energy

Kinetic energy is the energy a body possesses by virtue of being in motion. For electrons, this motion is often a result of acceleration through an electric field. When an electron is accelerated by a potential difference (\( V \) ), it gains kinetic energy given by the equation:\[ \text{Kinetic Energy} = \frac{1}{2} m v^2 = eV \]Here, \( e \) stands for the charge of the electron, and \( V \) is the potential difference it has been accelerated through. This equation indicates that the kinetic energy of an electron is directly proportional to the potential difference it experiences.Understanding this relationship is essential for determining how the electron’s speed and thus momentum increase. Higher kinetic energy means that electrons have more motion, increasing their momentum, and thereby affecting the computed de Broglie wavelength.

Potential Difference

Potential difference is a measure of how much energy is provided to a charged particle. It is essentially an electric potential that drives the movement of electrons. When electrons pass between two charged plates with a given potential difference (\( V \) ), they gain kinetic energy, enhancing their speed and motion. Potential difference, measured in volts, can be thought of as an energy shift enabling electrons to move from a lower energy state to a higher one. The relationship between potential difference and energy imparted to electrons is critical:

• The energy gained by each electron (in electron volts) is equivalent to the potential difference they are subjected to.
• This gained energy is what increases the electron's kinetic energy and, consequently, its momentum.

In the study of de Broglie wavelengths, understanding potential difference provides insight into how energy changes affect an electron’s behavior and characteristics, such as its associated wavelength.