In the operating
Room, anesthesiologists use mass spectrometers to monitor the respiratory gases of patients
undergoing surgery. One gas that is often monitored is the anesthetic isoflourane (molecular
mass = 3.06 · 10−25 kg). In a spectrometer, a singly ionized molecule of isoflourane (charge =
+e) moves at a speed of 7.2 · 103 m/s on a circular path that has a radius of 0.10 meters. What
is the magnitude of the magnetic field that the spectrometer uses?
Look up the formula for the 'Larmor' or 'cyclotron' radius of a particle in a magnetic field. It depends upon q, v and B.
It will let you solve for B.
If you can't find it in your textbook or derive it (which isn't hard), do a Google search. It will be a useful learning experience.
For a singly ionized molecule, q = e.
Hint: q V B = M V^2/R
To calculate the magnitude of the magnetic field that the spectrometer uses, we can use the formula for the magnetic force on a charged particle moving in a magnetic field.
The formula for the magnetic force (F) on a charged particle moving in a magnetic field is given by:
F = q * v * B * sin(θ)
Where:
- q is the charge of the particle (in coulombs),
- v is the velocity of the particle (in m/s),
- B is the magnitude of the magnetic field (in teslas), and
- θ is the angle between the velocity vector and the magnetic field vector.
In this case, the particle is a singly ionized molecule of isoflurane, with a charge of +e (elementary charge of 1.6 x 10^-19 C). The velocity of the particle is given as 7.2 x 10^3 m/s. The molecule moves in a circular path, so the angle between the velocity vector and the magnetic field vector is 90 degrees.
Plugging in these values, we can rearrange the formula to solve for the magnetic field (B):
B = F / (q * v * sin(θ))
Since the particle is moving in a circular path, the magnetic force provides the necessary centripetal force:
F = m * a
Where:
- m is the mass of the particle (in kg), and
- a is the centripetal acceleration (in m/s^2).
The centripetal acceleration is given by:
a = v^2 / r
Where:
- r is the radius of the circular path (in meters).
Plugging this expression for a into the equation for F:
F = m * (v^2 / r)
Now, substituting this expression for F back into the equation for B:
B = (m * v^2) / (q * v * r * sin(θ))
We can simplify this further:
B = m * v / (q * r * sin(θ))
Now, we can substitute the given values into this equation to find the magnitude of the magnetic field. The molecular mass of isoflurane is given as 3.06 x 10^-25 kg, the velocity is 7.2 x 10^3 m/s, the charge is +e (1.6 x 10^-19 C), the radius is 0.10 meters, and the sin(θ) is sin(90 degrees) = 1.
B = (3.06 x 10^-25 kg * 7.2 x 10^3 m/s) / (1.6 x 10^-19 C * 0.10 m * 1)
Now, calculate the value:
B = 1.08 x 10^-3 T
Therefore, the magnitude of the magnetic field that the spectrometer uses is approximately 1.08 x 10^-3 Tesla.