To determine the kinetic energy of the proton that describes a circular orbit of the same radius in a magnetic field, we need to consider the relationship between the mass of the particle, its charge, and the magnetic field strength. The key concept here is that the centripetal force required for circular motion in a magnetic field is provided by the magnetic Lorentz force.
Understanding the Forces at Play
When a charged particle moves in a magnetic field, it experiences a force that acts perpendicular to both its velocity and the magnetic field direction. This force can be expressed as:
Where:
- F is the magnetic force,
- q is the charge of the particle,
- v is the velocity of the particle, and
- B is the magnetic field strength.
Centripetal Force and Circular Motion
The centripetal force required to keep a particle in circular motion is given by:
Where:
- m is the mass of the particle,
- v is the velocity, and
- r is the radius of the circular path.
Equating Forces
For a charged particle in a magnetic field, we can set the magnetic force equal to the centripetal force:
Rearranging this equation gives us:
Kinetic Energy Relation
The kinetic energy (KE) of a particle is given by:
Substituting the expression for velocity from our earlier equation into the kinetic energy formula, we get:
- KE = 0.5m(qBr/m)² = 0.5(q²B²r²)/m
Comparing the Deuteron and Proton
Now, let's apply this to our specific case. The deuteron has a charge of +1e and a mass of approximately 2mp (where mp is the mass of a proton). The proton also has a charge of +1e but a mass of mp.
For the deuteron:
For the proton:
Relating the Energies
From the equations, we can see that:
Given that the kinetic energy of the deuteron is 50 keV, we can find the kinetic energy of the proton:
- KEp = 2 * 50 keV = 100 keV
Final Answer
Thus, the kinetic energy of the proton that describes a circular orbit of radius 0.5 m in the same magnetic field is 100 keV. Therefore, the correct answer is option 4) 100 keV.
