Final answer:
The proton has a greater de Broglie wavelength compared to the alpha particle when both are accelerated through the same potential because it has a lower mass. Both particles have the same kinetic energy as the potential difference imparts equal energy to each charge in the particles.
Step-by-step explanation:
When a proton and an alpha particle are accelerated through the same potential, we must consider their charge and mass to determine their de Broglie wavelength and kinetic energy. The de Broglie wavelength (λ) of a particle is inversely proportional to its momentum (p), which can be defined as λ = h/p, where h is Planck's constant. Since both particles are accelerated through the same potential, they gain the same amount of electrical energy which becomes their kinetic energy (KE). However, an alpha particle has a charge of +2e (where e is the elementary charge), and a mass approximately four times that of a proton.
(i) De Broglie wavelength: The proton has a greater value of de Broglie wavelength. This is because both particles have the same kinetic energy, but the alpha particle has a greater mass, leading to a higher momentum and thus a shorter de Broglie wavelength compared to a proton.
(ii) Kinetic energy: Since they are both accelerated through the same potential, both particles will have the same kinetic energy. The kinetic energy gained by each particle is equal to the charge of the particle multiplied by the potential difference. Despite the alpha particle having twice the charge of a proton, it also requires twice the work to be done against the electric field due to its double charge, leading to the same kinetic energy as that of the proton.