Charged particles in magnetic fields

Ions orbit Jupiter in a ring called a plasma torus as shown in Figure 6. This question is about applying gravitational, magnetic and electric fields to a particle in the plasma.

An ion travels in a circular path at a speed of 74.2 km s⁻¹ and at a distance of 4.22 × 10⁸ m from the centre of Jupiter.

Student A suggests that the gravitational force acting on the ion due to Jupiter provides the centripetal acceleration of the ion.

Deduce whether this suggestion is correct.

Student B suggests that the magnetic field of Jupiter alone can cause the ion to move in the circular path.

Figure 7 shows the direction of the velocity v of the ion in the plasma torus. v is tangential to the plasma torus.

Figure 7

The ion has a positive charge.

Student B ’s suggestion requires a magnetic field to act at the position of the ion.

Draw an arrow on Figure 7 to show the direction of this magnetic field.

The ion has a charge of +3.20 × 10⁻¹⁹ C and a mass of 5.31 × 10⁻²⁶ kg.

The ion moves with a speed of 74.2 km s⁻¹ in a circle of radius 4.22 × 10⁸ m.

Calculate the magnitude of the magnetic flux density required for this motion. State, in fundamental (base) units, the unit for your answer. Ignore any contribution from the gravitational field.

The plasma torus contains two ions X and Y as shown in Figure 8. X has a charge of +2 e. Y has a charge of +1 e. Assume that each ion acts as a point charge.

Calculate the magnitude of the resultant electric field strength at point P in Figure 8 due to the two ions.

An electron enters a uniform magnetic field at right angles to the field. The flux density of the field is B. The electron moves with a non-relativistic speed v in a circular path of radius r.

What is the number of circuits completed by the electron in one second?

A beam consists of ionised atoms of two isotopes of an element. When the beam enters a uniform magnetic field, the ions move in circular paths. The ions have the same charge and travel at the same speed when they enter the magnetic field.

Which statement is true?

When an electron moves at a speed v perpendicular to a uniform magnetic field of flux density B, the radius of its path is R. A second electron moves at a speed perpendicular to a uniform magnetic field of flux density 4 B.

What is the radius of the path of the second electron?

Figure 9 shows a cyclotron. A proton is released from rest and is accelerated each time it reaches the gap between two horizontal ‘dees’ D₁ and D₂. Between these accelerations the proton moves at constant speed. A vertical magnetic field of flux density B acts over the dees so that the proton follows a semicircular path in each dee.

The dees are connected to an alternating potential difference (pd). This pd is adjusted so that the proton is always accelerated by the peak electric field as it crosses the gap between the dees.

Explain why the proton travels in a semicircular path in a dee.

The peak pd of the alternating supply is 10.0 kV. The proton leaves the cyclotron with kinetic energy of 14 MeV.

Determine the number of times the proton moves across the gap before it leaves the cyclotron.

The radius of the outermost semicircular path of the proton is R and the proton leaves with a maximum kinetic energy E_k.

Show that E_k is given by

A hospital decides to purchase a cyclotron in order to manufacture its own radioactive isotopes using high-speed protons. The required minimum kinetic energy of the emerging protons is 11 MeV. The cost of a cyclotron is approximately proportional to E_k^1.5. The cost of a 10 MeV cyclotron is about £2.3 million. Table 1 gives information for three cyclotrons X, Y and Z.

Deduce which cyclotron X, Y or Z will satisfy the energy requirement for the lowest cost. Go on to determine the approximate cost of this cyclotron.

Mass spectrometers are used to measure the masses of ions.

Figure 3 shows one part of a mass spectrometer.

A narrow beam consists of positive lithium ions travelling at different speeds. The beam enters a region where there is an electric field and a magnetic field. The directions of the uniform electric field of strength E and the uniform magnetic field of flux density B are shown on Figure 3.

Most ions are deflected from their original path. Lithium ions that travel at one particular speed are not deflected, and pass through the small aperture.

The positive lithium ion A in Figure 3 moves at a speed v.

Draw two labelled arrows on Figure 3 to show the directions of the electric force F_E and the magnetic force F_M acting on A.

Lithium ions travelling at 1.5 × 10⁵ m s⁻¹ pass through the small aperture.

Calculate E.

B = 0.12 T

Ions that pass through the small aperture enter a second uniform magnetic field of flux density B. Ions of different mass are separated because they follow different paths as shown in Figure 4.

Ions of mass m and charge q travelling at speed v follow a circular path in the uniform magnetic field.

Show that the radius r of the circular path is given by

The ions of different mass are deflected and strike the detector surface at different distances from the small aperture as shown in Figure 4.

A singly-charged lithium ion (⁶₃Li⁺) passes through the small aperture.

Calculate the distance between the small aperture and the point where this ion strikes the detector surface.

B = 0.12 T

Figure 5 shows a different type of mass spectrometer working with lithium ions.

A stationary⁷₃Li⁺ ion in the lithium sample is at the mid-point between the parallel electrodes. The⁷₃Li⁺ ion accelerates towards aperture P.

Determine the speed of the ion when it emerges through aperture P.

⁶Li⁺ Li⁷⁺ L ₃ and₃ ions are produced in the sample simultaneously and travel a distance from aperture P to the detector. For each type of ion, the time interval between production and detection is measured.

Discuss how the masses of the ions can be deduced from the measurement of these time intervals.

When an electron is moving at a speed v perpendicular to a uniform magnetic field of flux density B, it follows a path of radius R. A second electron moves at a speed perpendicular to a uniform magnetic field of flux density 4 B.

What is the radius of the path of the second electron?

Different magnetic fields are present in the two chambers shown. A particle enters the first chamber at a velocity of 80 m s^–1 and is deflected into a circular path of radius 200 mm In the second chamber it follows a circular path of radius 100 mm

The particle leaves the second chamber at a speed of

An alternating electric field between the D-shaped regions accelerates charged particles. The magnetic field causes the charged particles to follow a circular path.

Figure 4 shows the path followed by a proton that starts from O.

Explain why it is not possible for the magnetic field to alter the speed of a proton while it is in one of the D-shaped regions.

Derive an expression to show that the time taken by a proton to travel round one semi-circular path is independent of the radius of the path.

The maximum radius of the path followed by the proton is 0.55 m and the magnetic flux density of the uniform field is 0.44 T.

Ignore any relativistic effects.

Calculate the maximum speed of a proton when it leaves the cyclotron.