Final answer:
The distance between the impact points formed on the photographic plate by singly charged ions of uranium-235 and uranium-238 in a mass spectrometer is approximately 1.791 x 10^(-6) cm.
Step-by-step explanation:
The motion of charged particles in a magnetic field is described by the Lorentz force equation:
F=qvB
where:
F is the magnetic force,
q is the charge of the particle,
v is the velocity of the particle,
B is the magnetic field strength.
The magnetic force provides the centripetal force for the circular motion of the ions. The centripetal force (Fc) is given by:
Fc = mv2/r
where:
m is the mass of the particle,
v is the velocity of the particle,
r is the radius of the circular path.
Setting the magnetic force equal to the centripetal force, we get:
qvB= mv2/r
Solving for
r, the radius of the circular path, we find:
r= mv/qB
Now, the distance traveled by the ion on the photographic plate (d) is related to the circular path by the formula:
d=2πr
Substituting the expression for
r into the formula for d, we get:
d=2π mv/qB
Now, let's calculate the distance d for both 235U and 238U ions. The masses and charges for 235U and 238U are different, so we need to take those values into account.
For 235U:
Mass (m235):
235×1.6605×10−27kg (mass of a proton)
Charge (q235): 1×e (where e is the elementary charge, approximately 1.602×10−19C)
For 238U:
Mass (m238):
238×1.6605×10−27kg
Charge (q238): 1×e
Now, we can substitute these values into the formula for d:
d = 2π m235v/ q238B
d' = 2π m235v/ q238B
Now, plug in the given values:
v=2.75×105m/s
B=0.580 T
Calculate d and d′, and the difference between them will give you the distance between the impact points on the photographic plate. Note that the masses are very close, so the difference in distances should be small.