Question

J. J. Thomson is best known for his discoveries about the nature of cathode rays. Another important contribution of his was the invention, together with one of his students, of the mass spectrometer. The ratio of mass m to (positive) charge q of an ion may be accurately determined in a mass spectrometer. In essence, the spectrometer consists of two regions: one that accelerates the ion through a potential difference V and a second that measures its radius of curvature in a perpendicular magnetic field. (Figure 1) The ion begins at potential V and is accelerated toward zero potential. When the particle exits the region with the electric field it will have obtained a speed u. Part A With what speed u does the ion exit the acceleration region? Find the speed in terms of m, q, V, and any constants. Part B After being accelerated, the particle enters a uniform magnetic field of strength B0 and travels in a circle of radius R (determined by observing where it hits on a screen--as shown in the figure). The results of this experiment allow one to findm/q in terms of the experimentally measured quantities such as the particle radius, the magnetic field, and the ap

Answer 1

Part A: The speed (u) with which the ion exits the acceleration region in the mass spectrometer can be determined using the conservation of energy. The kinetic energy gained by the ion is equal to the potential energy it loses during acceleration. The formula for the speed (u) is given by:

�

=

2

�

�

�

u=

m

2qV

​

​

where

�

q is the charge of the ion,

�

V is the potential difference, and

�

m is the mass of the ion.

Part B: After being accelerated, the ion enters a magnetic field and travels in a circle. The centripetal force is provided by the magnetic force, leading to the equation:

�

�

2

�

=

�

�

�

0

R

mv

2

​

=qvB

0

​

where

�

R is the radius of the circle,

�

0

B

0

​

is the magnetic field strength,

�

m is the mass of the ion,

�

q is the charge of the ion, and

�

v is the speed of the ion in the circular path.

By rearranging the terms, the ratio of mass to charge (

�

�

q

m

​

) can be expressed in terms of experimentally measured quantities:

�

�

=

�

�

0

q

m

​

=

B

0

​

R

​

Step-by-step explanation:

Part A: The expression for the speed (

�

u) comes from the conservation of energy, where the gained kinetic energy equals the lost potential energy during acceleration.

Part B: The equation for the ratio of mass to charge (

�

�

q

m

​

) is derived from the centripetal force in the magnetic field, expressing it in terms of the experimentally measurable quantities.

Answer 2

J. J. Thomson is known for his discoveries about the nature of cathode rays and the invention of the mass spectrometer. In a mass spectrometer, the ion is accelerated through a potential difference and its radius of curvature is measured in a magnetic field. The speed of the ion can be found using the equation u = sqrt(2qV/m), and the mass-to-charge ratio can be found using the equation m/q = (2V)/(B0^2 * R).

Step-by-step explanation:

J. J. Thomson is best known for his discoveries about the nature of cathode rays. One of his important contributions was the invention of the mass spectrometer together with one of his students. The mass spectrometer is a device that accurately determines the ratio of mass (m) to (positive) charge (q) of an ion. In a mass spectrometer, the ion is accelerated through a potential difference (V) and then its radius of curvature is measured in a perpendicular magnetic field.

Part A:

The ion exits the acceleration region with a speed (u), which can be found in terms of m, q, V, and any constants. The speed of the ion can be determined using the equation: u = sqrt(2qV/m).

Part B:

The particle then enters a uniform magnetic field (B0) and travels in a circle of radius (R). The mass-to-charge ratio (m/q) of the particle can be found in terms of experimentally measured quantities such as the particle radius, the magnetic field, and the acceleration voltage. The equation for finding m/q is: m/q = (2V)/(B0^2 * R).