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The potential difference plays a crucial role in the motion of charged particles, like protons and alpha particles. When a charged particle is subjected to a potential difference, it experiences a change in electric potential energy. This change can accelerate the particle. The amount of energy gained by the particle is known as its kinetic energy, which can be expressed as:

• For any particle with charge \( q \), the kinetic energy \( K \) gained is \( K = qV \), where \( V \) is the potential difference.

In our scenario, a proton with charge \( e \), and an alpha particle with charge \( 2e \), are both accelerated by the same potential difference. Despite being subjected to the same potential difference, different particles acquire different amounts of kinetic energy due to their unique charges. Understanding this concept helps us realize how kinetic energy and momentum relate to an electric potential which paves the way to uncovering deeper insights into particle physics.

An alpha particle, often symbolized as \( \alpha \), is a type of atomic particle found commonly in nuclear physics. It consists of two protons and two neutrons, making it a helium nucleus. Here are some key characteristics of alpha particles that are relevant:

• Charge: Being composed of two protons, an alpha particle has a total charge of \( +2e \), double that of a single proton.
• Mass: The mass of an alpha particle is roughly four times that of a single proton.
• Role in de-Broglie Wavelength: When calculating the de-Broglie wavelength of an alpha particle, its larger mass and double charge compared to a proton must be accounted for. This affects its momentum and ultimately its wavelength.

Recognizing these properties is essential when comparing the behaviors of alpha particles and protons, especially under the influence of external forces, such as an electric field caused by a potential difference.

Kinetic energy is the energy a particle has due to its motion. For charged particles accelerated by a potential difference, kinetic energy can be calculated directly from the charge and potential difference.

• Formula: The kinetic energy \( K \) gained by a charged particle is given by \( K = qV \), where \( q \) is the charge and \( V \) is the potential difference.
• Impact on de-Broglie Wavelength: As kinetic energy increases, a particle's momentum increases, and this reduces its de-Broglie wavelength, following the relationship \( \lambda = \frac{h}{p} \), where \( p \) is momentum.
• Comparison: When a proton and an alpha particle are accelerated through the same potential difference, the alpha particle gains more kinetic energy due to its greater charge \( 2e \), potentially resulting in a different wavelength compared to the proton.

This concept of kinetic energy is pivotal in linking electric fields to physical movement, allowing us to delve into the quantum behaviors of particles.

Momentum is a key ingredient in determining the de-Broglie wavelength of a particle. The momentum \( p \) of a particle is defined by its mass and velocity, and it helps in calculating the wavelength through the relation \( \lambda = \frac{h}{p} \), where \( h \) is Planck's constant.

• Relation to Kinetic Energy: For a moving particle, \( p = \sqrt{2mK} \), where \( m \) is the mass and \( K \) the kinetic energy.
• Calculating for Proton and Alpha Particle: In the exercise, the momentum for a proton and an alpha particle are given by their respective masses and the energy gained from the applied potential difference.
• Impact on Wavelength: Higher momentum results in shorter wavelengths, illustrating the wave-like property of particles, crucial to quantum mechanics.

Understanding the momentum formula provides a bridge between classical mechanics and quantum mechanics, manifesting in the particle-wave duality through de-Broglie's hypothesis.