A mass spectrometer is used to separate isotopes. If the bean emerges with a speed of 374 km/s and the magnetic field in the mass selector is 1.5 T, what is the distance between the collectors for 235U+1 and 238U+1? The mass of 235U+1 is 235 u and the mass of 238U+1 is 238 u, where the atomic mass unit u = 1.66 x 10-27 kg.
You seem to have omitted two important dimensions:
(1) the path length of the particle through the mass spectrometer's B field, and
(2) the distance to where the collectors are located. It is probably longer than the first dimension.
To solve this problem, we can use the principles of circular motion in a magnetic field.
First, let's determine the equation that relates the magnetic force and the centripetal force acting on a moving charged particle in a magnetic field:
F+ = qvB,
where F+ is the force on the positive ion, q is the charge of the ion, v is the velocity of the ion, and B is the magnetic field strength.
By equating this force to the centripetal force, we can find the radius of the circular path:
F+ = mω^2r,
where m is the mass of the ion and ω is the angular velocity of the ion.
We know that the centripetal force (F+) is equal to the Lorentz force (qvB), so we can equate them:
qvB = mω^2r.
We can rewrite ω in terms of the velocity and radius:
ω = v/r.
Substituting this into the equation above, we have:
qvB = mv^2/r.
Rearranging the equation, we get:
r = mv / (qB).
Now let's apply this equation to calculate the radii for the two ions, 235U+1 and 238U+1.
For 235U+1:
- Mass (m) = 235 u = 235 × 1.66 x 10^-27 kg
- Charge (q) = +1 (as indicated by "+1")
- Velocity (v) = 374 km/s = 374 x 1000 m/s
- Magnetic field (B) = 1.5 T
Substitute these values into the equation:
r(235) = (235 × 1.66 x 10^-27 kg) × (374 x 1000 m/s) / (+1) × (1.5 T).
Calculate the value to find the radius (r) for the 235U+1 ion.
Repeat the same steps for the 238U+1 ion using the mass of 238 u.
Finally, to find the distance between the collectors, subtract the radii of the two ions:
Distance = r(238) - r(235).