To determine the energy of an α particle that would allow it to move in a circular path of the same radius R in a uniform magnetic field as a proton with 1 MeV kinetic energy, we need to consider the relationship between kinetic energy, charge, mass, and the radius of the circular motion in a magnetic field.
Key Concepts
- Kinetic Energy (KE): The energy an object possesses due to its motion, given by the formula KE = (1/2)mv².
- Magnetic Force: The force acting on a charged particle moving in a magnetic field, which causes circular motion.
- Radius of Circular Motion: For a charged particle in a magnetic field, the radius R is given by R = (mv)/(qB), where m is mass, v is velocity, q is charge, and B is the magnetic field strength.
Comparing Proton and Alpha Particle
A proton has a charge of +1e and a mass of approximately 1 u (atomic mass unit). An α particle consists of 2 protons and 2 neutrons, giving it a charge of +2e and a mass of about 4 u.
Finding the Required Energy
Since both particles are moving in the same magnetic field and describe a circle of the same radius, we can set up the relationship:
For the proton:
R = (mv)/(qB)
For the α particle:
R = (M*V)/(QB)
Where:
- m = mass of the proton
- M = mass of the α particle
- v = velocity of the proton
- V = velocity of the α particle
- q = charge of the proton
- Q = charge of the α particle
Since the radius R is the same, we can equate the two expressions. The α particle's mass is four times that of the proton, and its charge is twice that of the proton. Therefore, to maintain the same radius, the kinetic energy of the α particle must be:
KE(α) = 4 * KE(proton) = 4 * 1 MeV = 4 MeV.
Final Answer
The energy of the α particle required to describe a circle of the same radius in the same magnetic field is 4 MeV.
