An electron with a kinetic energy of 0.8 keV is fired normal to the lines of a field of magnitude 50 G. Find:
a)the radius of the path
b)its acceleration
c)its period
To find the answers to these questions, we can use the principles of the Lorentz force. The Lorentz force is the force experienced by a charged particle moving through a magnetic field. It is given by the equation:
F = q(v x B),
where F is the force, q is the charge of the particle, v is the velocity of the particle, and B is the magnetic field vector.
a) To find the radius of the path, we need to use the equation for the centripetal force acting on the electron:
F = (m*v^2)/r,
where F is the centripetal force, m is the mass of the electron, v is the velocity of the electron, and r is the radius of the path.
Since the Lorentz force is acting as the centripetal force, we can set these two equations equal to each other:
q(v x B) = (m*v^2)/r.
The speed of the electron can be calculated using the kinetic energy:
KE = (1/2)m*v^2.
Since the kinetic energy is given as 0.8 keV, we convert it to joules:
KE = 0.8 * 1.6e-19 J.
We can rearrange the kinetic energy equation to solve for v:
v^2 = (2*KE)/m.
Now, we substitute v^2 and rearrange the equation to solve for r:
q(v x B) = (m*v^2)/r,
qvB = (m*2*KE)/r,
r = (m*v)/(q*B).
Inserting the given values, the mass of an electron (m) is 9.11e-31 kg, the charge of an electron (q) is -1.6e-19 C, the kinetic energy (KE) is 0.8 * 1.6e-19 J, and the magnetic field (B) is 50 G (or 50e-4 T):
r = (9.11e-31 * sqrt((2*0.8*1.6e-19)/(9.11e-31))) / (-1.6e-19 * 50e-4).
Solving this equation will give you the radius of the path of the electron.
b) The acceleration of the electron can be found using the equation:
F = q(v x B),
where F is the force acting on the electron due to the magnetic field. Since the force is acting as centripetal force, we can use the equation:
F = m*a,
where m is the mass of the electron, and a is the acceleration of the electron.
Equating the two equations, we have:
q(v x B) = m*a.
Rearranging the equation, we can solve for acceleration:
a = (q/m) * (v x B).
Using the given values, you can substitute to find the acceleration of the electron.
c) The period can be found using the equation:
T = 2π/ω,
where T is the period, and ω is the angular velocity.
The angular velocity can be calculated using the equation:
ω = v/r.
Substituting the values of v and r from the previous calculations, you can solve for the period.