Problem 53 - Master Resource Book in JEE Main Physics

Question

An electron of mass \(m\) and charge \(e\) initially at rest gets accelerated by a constant electric field \(E\). The rate of change of de-Broglie wavelength of this electron at time \(t\) ignoring relativistic effect is (a) \(\frac{-h}{e E t^{2}}\) (b) \(\frac{-e E t}{E}\) (c) \(\frac{-m h}{e E t^{2}}\) (d) \(\frac{-h}{e E}\)

Step-by-Step Solution

Answer

Option (a) \( \frac{-h}{eEt^2} \) is correct.

Step 1: Understand the de-Broglie Wavelength

The de-Broglie wavelength \( \lambda \) of a particle is given by \( \lambda = \frac{h}{p} \), where \( h \) is Planck's constant and \( p \) is the momentum of the particle. For an electron, we need its momentum in terms of the given parameters to find the rate of change of this wavelength.

Step 2: Determine the Momentum of the Electron

Since the electron is accelerated by a constant electric field \( E \), the force on the electron is \( F = eE \). Using Newton's second law, \( F = ma \), the acceleration \( a = \frac{eE}{m} \). The velocity at time \( t \) is \( v = at = \frac{eEt}{m} \). Therefore, the momentum \( p = mv = m \cdot \frac{eEt}{m} = eEt \).

Step 3: Differentiate the de-Broglie Wavelength with Respect to Time

Substitute the expression for momentum \( p = eEt \) into the de-Broglie wavelength formula to get \( \lambda = \frac{h}{eEt} \). Differentiate this with respect to \( t \):\( \frac{d\lambda}{dt} = \frac{d}{dt} \left( \frac{h}{eEt} \right) = \frac{h}{eE} \cdot \left(-\frac{1}{t^2}\right) = \frac{-h}{eEt^2} \).

Step 4: Compare with Given Options

The calculated rate of change of de-Broglie wavelength is \( \frac{-h}{eEt^2} \). Compare this with the given options. Option (a) \( \frac{-h}{eEt^2} \) matches the result.

Key Concepts

Electron Momentum

In physics, momentum is a crucial concept that helps describe the motion of objects. For an electron, which is a charged particle, momentum is defined as the product of its mass and its velocity. In this context, since the electron is being accelerated by an electric field, its momentum will change over time. The equation for momentum is given by:

• Momentum (\( p \)) = mass (\( m \)) * velocity (\( v \))

By using Newton's second law, we can determine the force on the electron as the product of its charge (\( e \)) and the electric field (\( E \)), which allows us to find its momentum. The formula becomes:

• Momentum (\( p \)) = charge (\( e \)) * electric field (\( E \)) * time (\( t \))

This relationship tells us that as time progresses, the momentum of the electron increases linearly, which is essential for understanding the rate of change of its de-Broglie wavelength.

Electric Field

An electric field is a region around a charged object where other charged objects experience a force. The strength and direction of this field can affect how charged particles, like electrons, move. When an electron is within an electric field, it experiences a force that accelerates it, which changes its velocity over time. This force is calculated using the equation:

• Force (\( F \)) = charge (\( e \)) * electric field (\( E \))

The electric field causes the electron to gain speed and therefore momentum, altering its kinetic energy. This is a pivotal part of understanding how the de-Broglie wavelength changes, as it directly influences the momentum of the electron. By applying a constant electric field, the particle's velocity and hence its momentum increases linearly with time.

Rate of Change

The rate of change in physics refers to how quickly a quantity, like velocity or momentum, changes with time. For the de-Broglie wavelength, which relates a particle's momentum to its wave characteristics, this rate is crucial. The equation that links the de-Broglie wavelength to momentum is:

• De-Broglie wavelength (\( \lambda \)) = \( \frac{h}{p} \)

where \( h \) is Planck's constant and \( p \) is the momentum. As momentum changes with time, so does the wavelength. By differentiating the de-Broglie wavelength with respect to time, we can determine how quickly it changes, known as its rate of change. This calculation gives insight into wave-particle duality and other fundamental principles of quantum mechanics.

Planck's Constant

Planck's constant (\( h \)) is a fundamental number in quantum mechanics, and it's crucial for linking energy and frequency in wave-particle interactions. Its value is approximately \( 6.626 imes 10^{-34} \ ext{Js} \). When applied to the de-Broglie wavelength equation, Planck's constant serves as a bridge between the wave nature and particle nature of electrons.

• De-Broglie wavelength (\( \lambda \)) equation: \( \lambda = \frac{h}{p} \)

This shows that an electron's wave-like behavior is inversely proportional to its momentum. In our original exercise, Planck's constant remains the same because it is a universal constant, but it profoundly affects how changes in momentum translate to changes in wavelength. Understanding Planck's constant is essential for anyone studying quantum mechanics and the fundamental behavior of particles.