Consider a mass spectrometer like the one shown in the figure. A beam of singly charged positive ions is accelerated from an ion source and injected into a velocity selector with an electric field of E=2700V/m) and magnetic field of B=43.0mT. The ions then enter the deflection region where a uniform B0=39.0mT magnetic field is maintained. What is the mass of these ions if they are observed to strike a detector at a distance of 0.686m from the entrance of the deflection chamber? Answer in units of kg.
To determine the mass of the ions, we can use the equation for the velocity of a charged particle in a magnetic field.
First, let's calculate the velocity of the ions when entering the deflection region. We can use the velocity selector, which acts as a filter to allow only ions with a specific velocity to pass through. In this case, the ions are singly charged, so we can consider the electric field in the velocity selector.
The electric force experienced by a charged particle can be calculated using the equation:
F = q * E,
where F is the electric force, q is the charge of the particle, and E is the electric field strength. In this case, the charge of the ions is +1e (since they are singly charged positive ions). Therefore, the electric force can be written as:
F = (1e) * E,
where e is the elementary charge (e = 1.6022 × 10^(-19) C).
The electric force is balanced by the magnetic force, which is given by the equation:
F = q * v * B,
where F is the magnetic force, q is the charge of the particle, v is the velocity of the particle, and B is the magnetic field strength. Rearranging this equation, we can solve for the velocity:
v = F / (q * B).
Now, let's calculate the velocity of the ions at the entrance of the deflection chamber. The force experienced by the ions in the electric field of the velocity selector is given by:
F = (1e) * E = (1e) * (2700 V/m).
Substituting these values into the equation for the velocity, we have:
v = [(1e) * (2700 V/m)] / [(1e) * (43.0 mT)].
Simplifying the equation, we get:
v = (2700 V/m) / (43.0 mT).
Now, we can calculate the velocity:
v = (2700/43) m/s.
Next, let's determine the radius of the ion's circular path in the deflection chamber. The magnetic force acting on the ion in the deflection chamber provides the centripetal force required for the circular motion. It can be calculated using the equation:
F = q * v * B0,
where F is the magnetic force, q is the charge of the particle, v is the velocity of the particle, and B0 is the magnetic field strength in the deflection chamber.
The magnetic force is equal to the centripetal force, which can be written as:
F = (m * v^2) / r,
where m is the mass of the ion and r is the radius of the circular path.
Equating the magnetic force and the centripetal force, we have:
(q * v * B0) = (m * v^2) / r.
Rearranging this equation, we can solve for the radius of the circular path:
r = (m * v) / (q * B0).
Now, let's calculate the radius:
r = (m * v) / [(1e) * (39.0 mT)].
The velocity (v) and magnetic field (B0) have been calculated earlier, so we can subtitute these values into the equation to get the radius (r) in terms of the mass (m).
Finally, we can determine the mass of the ions by equating the radius (r) to the given distance of the detector from the entrance of the deflection chamber. In this case, the distance is 0.686 m. So, we have:
0.686 m = (m * v) / [(1e) * (39.0 mT)].
Now, rearranging the equation, we can solve for the mass (m):
m = (0.686 m) * [(1e) * (39.0 mT)] / v.
Substitute the calculated values for distance, electric field, magnetic field, and the calculated velocity to find the mass of the ions in kg.