Problem 76 - Fundamentals of Physics

Question

Bainbridge's mass spectrometer, shown in Fig. 28-54, separates ions having the same velocity. The ions, after entering through slits, \(\mathrm{S}_{1}\) and \(\mathrm{S}_{2}\), pass through a velocity selector composed of an electric field produced by the charged plates \(P\) and \(P^{\prime},\) and a magnetic field \(\vec{B}\) perpendicular to the electric field and the ion path. The ions that then pass undeviated through the crossed \(\vec{E}\) and \(\vec{B}\) fields enter into a region where a second magnetic field \(\vec{B}^{\prime}\) exists, where they are made to follow circular paths. A photographic plate (or a modern detector) registers their arrival. Show that, for the ions, \(q / m=E / r B B^{\prime}\), where \(r\) is the radius of the circular orbit.

Step-by-Step Solution

Answer

\( \frac{q}{m} = \frac{E}{r B B'} \).

Step 1: Understanding Ion Velocity in the Selector

The devices work by allowing only ions with a specific velocity to pass through without deflection. The conditions for no deflection are defined by equating the electric force and the magnetic force acting on the ion: \( qE = qvB \). Simplifying, the expression for the velocity \(v\) of the ions that pass through the velocity selector is obtained: \( v = \frac{E}{B} \).

Step 2: Describe the Ion Trajectory in the Magnetic Field

Once the ions exit the velocity selector, they enter another magnetic field \( B' \), where they begin to move in a circular path due to the Lorentz force. In a magnetic field, the force on an ion is given by \( qvB' \), which acts as the centripetal force, \( \frac{mv^2}{r} \).

Step 3: Equate the Magnetic and Centripetal Forces

To find the relationship between charge and mass, set the magnetic force equal to the centripetal force: \[ q v B' = \frac{mv^2}{r} \]This equation is crucial as it relates the magnetic field to the motion characteristics of the ion.

Step 4: Substitute Ion Velocity into the Force Equation

Substitute the velocity \( v = \frac{E}{B} \) (found in Step 1) into the equation from Step 3: \[ q \left( \frac{E}{B} \right) B' = \frac{m (\frac{E}{B})^2}{r} \].

Step 5: Solve for the Charge-Mass Ratio (q/m)

Simplify the equation from Step 4 to solve for \( \frac{q}{m} \):\[ qE B' = \frac{m E^2}{r B^2} \]Dividing both sides by \( EB' \) gives:\[ \frac{q}{m} = \frac{E}{r B B'} \].

Step 6: Conclusion on the Charge-to-Mass Ratio

Thus, for ions passing through the mass spectrometer, the charge-to-mass ratio \( \frac{q}{m} \) is given by \( \frac{E}{rBB'} \). This final result is derived by balancing the forces and using characteristics of the motion in magnetic fields.

Key Concepts

Charge-to-Mass Ratio

In the context of a mass spectrometer, the charge-to-mass ratio \(\frac{q}{m}\) is an essential quantity that helps identify ions according to their mass properties. Understanding this ratio involves knowing how ions move through electric and magnetic fields. The charge-to-mass ratio indicates how much electric or magnetic force is required to produce a certain acceleration.
The equation \( \frac{q}{m} = \frac{E}{rBB'} \) can be derived by examining how forces affect ions within a spectrometer. This ratio is used to determine the unknown mass of ionized particles based on their known electric charge by tracking their path in a magnetic field.
The quantification of this ratio:

• Enables the identification and analysis of different ions.
• Is critical for applications like determining isotopic distributions or verifying molecular structures using mass spectrometry.

Through these principles, scientists can effectively identify particles by comparing their charge-to-mass ratios.

Velocity Selector

The velocity selector plays a crucial role in Bainbridge's mass spectrometer by ensuring that only ions with a specific velocity pass through a combination of electric and magnetic fields without deflection. The velocity selector operates using the balance between the electric force \( qE \) and the magnetic force \( qvB \), such that these forces are equal: \( qE = qvB \).
From this balance, we derive the expression for ion velocity: \( v = \frac{E}{B} \). The velocity selector effectively filters velocities, allowing only beams of ions with the desired speed to proceed further into the spectrometer.
Key functions of the velocity selector:

• Filters ions to ensure uniformity in velocity, crucial for accurate mass measurements.
• Enables precise interaction with subsequent magnetic fields, as all ions possess identical speeds once filtered.

Thus, the velocity selector is integral to achieving accurate mass spectrometry results by setting a standard velocity for further ion manipulation.

Magnetic Field

Magnetic fields in a Bainbridge mass spectrometer are pivotal in guiding ion paths. Within this device, ions encounter two distinct magnetic fields. Initially, there's a magnetic field aligned with the velocity selector to work alongside an electric field, ensuring ions pass through undeviated. Later, in the region following the velocity selector, a secondary magnetic field \( B' \) causes ions to follow circular paths due to the Lorentz force \( qvB' \).
The essential equations involving magnetic fields:

• Centripetal force provided by \( qvB' \) equals \( \frac{mv^2}{r} \), relating magnetic influence to circular motion.
• These fields manipulate ion trajectories for mass analysis, with the field's properties determining the radius \( r \) of these paths.

Ultimately, magnetic fields within a spectrometer are crucial for transforming linear ion paths into manageable circular ones. This concept allows for the differentiation and precise measurement of ions based on their mass and charge attributes.