An -particle has a charge of +2e and a mass of 6.64 10-27 kg. It is accelerated from rest through a potential difference that has a value of 1.10 106 V and then enters a uniform magnetic field whose magnitude is 3.10 T. The -particle moves perpendicular to the magnetic field at all times.
(a) What is the speed of the -particle?
m/s
(b) What is the magnitude of the magnetic force on it?
N
(c) What is the radius of its circular path?
m
To solve this problem, we can use the principles of electrical and magnetic forces and apply the concepts of potential difference, acceleration, and circular motion.
(a) To find the speed of the -particle, we can use the principle of conservation of energy. The potential energy gained by the particle when it moves through the potential difference is converted into kinetic energy. The equation relating potential energy to kinetic energy is:
ΔPE = ΔKE
The potential energy gained by the particle is given by:
ΔPE = q * ΔV
where q is the charge of the -particle (+2e) and ΔV is the potential difference (1.10 * 10^6 V).
The kinetic energy of the particle is given by:
ΔKE = (1/2) * m * v^2
where m is the mass of the -particle (6.64 * 10^-27 kg) and v is its velocity.
By equating ΔPE and ΔKE, we have:
q * ΔV = (1/2) * m * v^2
Rearranging this equation and solving for v, we get:
v = √(2 * q * ΔV / m)
Substituting the given values, we have:
v = √(2 * (2e) * (1.10 * 10^6 V) / (6.64 * 10^-27 kg))
Calculating this expression will give us the speed of the -particle.
(b) The magnetic force experienced by a charged particle moving through a magnetic field is given by the equation:
F = q * v * B
where q is the charge of the -particle (+2e), v is its velocity, and B is the magnitude of the magnetic field (3.10 T).
Substituting the given values, we can calculate the magnetic force on the -particle.
(c) The radius of the circular path can be determined by using the formula for the centripetal force:
F = (m * v^2) / r
where m is the mass of the -particle (6.64 * 10^-27 kg), v is its velocity, and r is the radius of the circular path.
Rearranging this equation for r, we have:
r = (m * v) / F
Substituting the known values, we can evaluate the radius of the circular path.
By following these steps, we can find the speed of the -particle, the magnitude of the magnetic force on it, and the radius of its circular path.