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Worksheet: Motion of a Charged Particle in a Magnetic Field

Q1:

A positive charge moves within regions containing magnetic fields, moving perpendicular to those fields. The moving charge is subject to magnetic forces, as shown in the cases (a), (b), and (c).

In what direction is the magnetic field aligned in case (a)?

  • Adown
  • Bright
  • Cout
  • Din
  • Eup

In what direction is the magnetic field aligned in case (b)?

  • Aleft
  • Bout
  • Cdown
  • Dright
  • Eup

In what direction is the magnetic field aligned in case (c)?

  • Aup
  • Bin
  • Cout
  • Dright
  • Edown

Q2:

A cosmic-ray electron moves at 7 . 5 × 1 0 / m s perpendicular to Earth’s magnetic field at an altitude where the field strength is 1 . 0 × 1 0 T . What is the radius of the circular path the electron follows? Electron mass is 9 . 1 × 1 0 kg and electron charge is 1 . 6 × 1 0 C.

Q3:

Two particles have the same linear momentum, but particle 𝐴 has four times the charge of particle 𝐵 . If both particles move in a plane perpendicular to a uniform magnetic field, what is the ratio 𝑅 𝑅 of the radii of their circular orbits?

  • A 1 8
  • B 1 1 6
  • C 1 2
  • D 1 4
  • E 1 1

Q4:

Find the radius of curvature of the circular path followed by a 25.0-MeV-energy proton that enters a region perpendicular to a 1.20-T magnetic field.

Q5:

A physicist is designing a cyclotron to accelerate protons to one-tenth the speed of light. The magnetic field will have a strength of 1.5 T.

Determine the rotational period of the circulating protons.

  • A 3 . 2 × 1 0 8 s
  • B 2 . 4 × 1 0 8 s
  • C 3 . 6 × 1 0 8 s
  • D 4 . 4 × 1 0 8 s
  • E 5 . 0 × 1 0 8 s

Determine the maximum radius of the protons’ orbit.

Q6:

A particle with charge-to-mass ratio 6 . 3 × 1 0 7 C/kg is in the atmosphere of a planet with no natural magnetic field. The particle is moving toward the center of the planet at a speed of 4 . 6 × 1 0 6 m/s as it enters an artificial uniform magnetic field of magnitude 0.75 T that is horizontal and directed from east to west.

What is the radius of the circular path that the particle follows while in the magnetic field?

What is the speed of the particle after it has moved in the field for 1 . 0 × 1 0 5 s?

  • A 4 . 6 × 1 0 6 m/s
  • B 1 . 4 × 1 0 6 m/s
  • C 9 . 5 × 1 0 6 m/s
  • D 7 . 2 × 1 0 6 m/s
  • E 6 . 5 × 1 0 6 m/s

Q7:

A physicist performing a sensitive measurement within Earth’s magnetic field wants to limit the magnetic force on a moving charged particle in her equipment to less than 2 . 1 3 × 1 0 1 2 N. What is the greatest magnitude of charge the particle can have if it moves at a maximum speed of 45.2 m/s? Take Earth’s magnetic field at the location of the experiment to be 25.0 µT.

  • A 2 . 0 3 × 1 0 9 C
  • B 2 . 1 0 × 1 0 9 C
  • C 1 . 6 0 × 1 0 9 C
  • D 1 . 8 8 × 1 0 9 C
  • E 1 . 9 2 × 1 0 9 C

Q8:

What magnitude of magnetic field is required to confine a proton moving with a speed of 6 . 3 × 1 0 6 m/s to a circular orbit of radius 15 cm?

Q9:

A uniform magnetic field of magnitude B is directed parallel to the 𝑧 -axis. A proton enters the field, moving with a velocity v j k = ( 3 . 0 + 4 . 0 ) × 1 0 6 m/s, and travels in a helical path with a radius of 4.8 cm.

What is the magnitude of B ?

What is the period of the particle’s motion along the 𝑥 -axis?

  • A 1 . 0 × 1 0 7 s
  • B 1 . 2 × 1 0 7 s
  • C 0 . 9 5 × 1 0 7 s
  • D 2 . 1 × 1 0 7 s
  • E 3 . 3 × 1 0 7 s

What is the distance of the particle from the point that it entered the field after it has moved in the field for 5 . 0 × 1 0 7 s?

Q10:

A negative charge moves within regions containing magnetic fields, moving perpendicular to those fields. The moving charge is subject to magnetic forces, as shown in the cases (a), (b), and (c).

In what direction does the negative charge move to produce the force on it that is shown in case (a)?

  • Aleft
  • Bup
  • Cdown
  • Dright
  • Ein

In what direction does the negative charge move to produce the force on it that is shown in case (b)?

  • Ain
  • Bright
  • Cleft
  • Ddown
  • Eout

In what direction does the negative charge move to produce the force on it that is shown in case (c)?

  • Ain
  • Bup
  • Cdown
  • Dleft
  • Eright

Q11:

An alpha-particle has a mass of 6 . 6 4 × 1 0 2 7 kg and a charge of 3 . 2 × 1 0 1 9 C. The alpha-particle travels in a circular path of radius 25 cm in a uniform magnetic field of magnitude 1.5 T.

What is the speed of the particle?

  • A 1 . 4 × 1 0 7 m/s
  • B 1 . 1 × 1 0 7 m/s
  • C 2 . 1 × 1 0 7 m/s
  • D 1 . 8 × 1 0 7 m/s
  • E 2 . 5 × 1 0 7 m/s

What is the kinetic energy of the particle?

  • A 6 . 8 × 1 0 6 eV
  • B 8 . 3 × 1 0 6 eV
  • C 6 . 0 × 1 0 6 eV
  • D 3 . 6 × 1 0 6 eV
  • E 9 . 0 × 1 0 6 eV

Through what potential difference must the particle be accelerated in order to follow the path that it moves along?

Q12:

A particle’s path is curved when it passes through a region of nonzero magnetic field, but its speed remains unchanged. This is very useful for beam steering in particle accelerators. A proton of speed 3 . 9 × 1 0 6 m/s enters a region of uniform magnetic field of magnitude 0.40 T that extends throughout a region of width 6.0 cm. The magnetic field direction is perpendicular to the velocity of the particle. Through what angle will the path of the proton change from the point that the particle enters the region containing the field to the point that it leaves the region?

Q13:

An ion has the same charge as an electron but a different mass. When moving in a magnetic field of magnitude 2 . 0 × 1 0 2 T, the ion takes 2.0 ms to complete exactly eight revolutions. Find the mass of the ion.

  • A 2 . 0 × 1 0 2 5 kg
  • B 1 . 6 × 1 0 2 5 kg
  • C 2 . 5 × 1 0 2 5 kg
  • D 1 . 3 × 1 0 2 5 kg
  • E 3 . 0 × 1 0 2 5 kg

Q14:

Viewers of Star Trek have heard of an antimatter drive on the Starship Enterprise. One possibility for such a futuristic energy source is to store charged antiparticles in a vacuum chamber using a magnetic field. The antiparticles can be extracted to annihilate them with normal particles in order to convert their masses to energy. What magnitude magnetic field is needed to maintain antiprotons moving at 7 . 3 0 × 1 0 7 m/s in a circular path of radius 3.75 m? Antiprotons have the same mass and magnitude of charge as protons, but negative rather than positive.

Q15:

A proton enters a uniform magnetic field ( 0 . 8 0 + 0 . 3 0 ) i k T with a velocity of ( 2 . 7 + 7 . 1 ) × 1 0 i j 6 m/s. What is the magnetic force on the proton?

  • A ( 3 . 4 , 1 . 3 , 9 . 1 ) × 1 0 i j k 1 3 N
  • B ( 3 . 4 , 1 . 3 , 9 . 1 ) × 1 0 i j k 1 3 N
  • C ( 1 . 3 , 9 . 1 , 3 . 4 ) × 1 0 i j k 1 3 N
  • D ( 3 . 4 , 1 . 3 , 9 . 1 ) × 1 0 i j k 1 3 N
  • E ( 9 . 1 , 1 . 3 , 4 . 2 ) × 1 0 i j k 1 3 N

Q16:

An oxygen-16 ion with a mass of 2 . 6 6 × 1 0 2 6 kg travels at 5 . 0 × 1 0 6 m/s perpendicular to a 1.20-T magnetic field. The ion moves in a circular arc with a 0.231-m radius.

What is the magnitude of the charge of the ion?

  • A 3 . 2 × 1 0 1 9 C
  • B 1 . 6 × 1 0 1 9 C
  • C 6 . 4 × 1 0 1 9 C
  • D 4 . 8 × 1 0 1 9 C
  • E 8 . 0 × 1 0 1 9 C

How many times greater is the magnitude of the charge of the ion than the magnitude of the charge of an electron?

Q17:

An electron (charge of 1 . 6 0 × 1 0 1 9 C) moving at 4 . 0 0 × 1 0 3 m/s in a 1.25-T magnetic field experiences a magnetic force of magnitude 1 . 4 0 × 1 0 1 6 N. The velocity of the electron can make two possible angles with the magnetic field. What is the value of the smaller angle?

Q18:

A cosmic ray proton moving toward Earth at a speed of 3 . 4 7 × 1 0 7 m/s experiences a magnetic force of 2 . 6 0 × 1 0 1 6 N from Earth’s magnetic field when at a point 𝑃 where the direction of the proton’s velocity makes an angle of 7 5 with the field. What is the magnitude of Earth’s magnetic field at point 𝑃 ?

  • A 6 . 2 3 × 1 0 5 T
  • B 3 . 1 1 × 1 0 5 T
  • C 5 . 1 2 × 1 0 5 T
  • D 4 . 8 5 × 1 0 5 T
  • E 4 . 9 0 × 1 0 5 T

Q19:

A singly charged particle is moving in a uniform magnetic field of magnitude 7 . 9 0 × 1 0 2 T and completes 13 revolutions in 3 . 4 7 × 1 0 4 s. What is the mass of the particle?

  • A 6 . 3 3 × 1 0 2 8 kg
  • B 9 . 1 4 × 1 0 2 8 kg
  • C 8 . 4 2 × 1 0 2 8 kg
  • D 5 . 3 7 × 1 0 2 8 kg
  • E 6 . 2 7 × 1 0 2 8 kg

Q20:

The magnitudes of the magnetic and electric fields in the velocity selector of a Bainbridge mass spectrometer are 𝐵 = 0 . 5 0 0 T and 𝐸 = 1 . 2 0 × 1 0 5 V/m, respectively, and the magnitude of the magnetic field that separates the ions emitted from the velocity selector is 0.750 T. A stream of singly charged Na ions is found to bend in a circular arc of radius 7.64 cm. What is the mass of the Na ions?

  • A 1 0 . 1 × 1 0 2 6 kg
  • B 4 . 9 6 × 1 0 2 6 kg
  • C 7 . 0 4 × 1 0 2 6 kg
  • D 3 . 8 2 × 1 0 2 6 kg
  • E 1 . 2 4 × 1 0 2 6 kg

Q21:

A water droplet has a mass of 1 . 0 × 1 0 4 g and an electric charge of 2 . 0 × 1 0 8 C. The water droplet is placed in a uniform magnetic field created in a laboratory. The uniform magnetic field points horizontally in the j direction, and the gravitational field of Earth points downwards in the 𝑘 direction. If the droplet is given an initial horizontal velocity of 5 . 0 × 1 0 5 i m/s, what must the magnitude of the magnetic field be to keep it moving in this direction? Use a value of 9.8 m/s2 for the acceleration due to gravity.

  • A 3 . 1 × 1 0 5 j T
  • B 1 . 6 × 1 0 5 j T
  • C 4 . 9 × 1 0 5 j T
  • D 9 . 8 × 1 0 5 j T
  • E 8 . 0 × 1 0 5 j T

Q22:

An alpha particle has a mass 𝑚 = 6 . 6 4 × 1 0 2 7 k g and a charge 𝑞 = 3 . 2 × 1 0 1 9 C . The alpha particle has a velocity v i k = ( 4 . 6 4 . 2 ) × 1 0 / 6 m s as it enters a region containing an electric field E i j = ( 2 . 5 4 . 0 ) × 1 0 / 4 V m and a magnetic field B i k = ( 1 . 0 + 1 . 0 ) × 1 0 2 T . What force acts on the alpha particle as it enters the field?

  • A ( 0 . 8 0 + 4 . 1 ) × 1 0 i j 1 4 N
  • B ( 0 . 8 0 4 . 1 ) × 1 0 i j 1 4 N
  • C ( 0 . 8 0 4 . 1 ) × 1 0 i k 1 4 N
  • D ( 0 . 8 0 4 . 1 ) × 1 0 i j 1 4 N
  • E ( 0 . 8 0 4 . 1 ) × 1 0 j k 1 4 N

Q23:

Calculate the magnetic force on a particle carrying a charge of 9 . 4 × 1 0 1 8 C and moving with a velocity of 5 . 4 × 1 0 4 i m/s in a magnetic field B k = 1 . 2 T.

  • A 1 . 2 × 1 0 1 3 j N
  • B 5 . 9 × 1 0 1 3 i N
  • C 7 . 4 × 1 0 1 3 k N
  • D 6 . 1 × 1 0 1 3 j N
  • E 3 . 3 × 1 0 1 3 j N

Q24:

A proton moves at 3 . 4 × 1 0 6 m/s in the positive 𝑦 -direction as it crosses the 𝑥 -axis at the position 𝑥 = 3 . 0 c m at an instant 𝑡 . There is a negatively charged particle 𝑄 fixed at the origin, and a uniform magnetic field B points in the positive 𝑧 -direction. The proton moves along a circular path of radius 3.0 cm about 𝑄 . If the charge 𝑄 is the same magnitude as the charge of the proton, what is the magnitude of B ?

Q25:

A proton, deuteron, and an alpha-particle are all accelerated through the same potential difference. These particles enter the same magnetic field, all of them moving perpendicular to the field. Assume that the mass of a deuteron is exactly twice that of a proton and the mass of an alpha-particle is exactly twice that of a deuteron.

What is the ratio of the radii of the circular paths followed by the proton and by the deuteron?

  • A 1 0 . 5
  • B 1 1
  • C 1 0 . 3 5
  • D 1 0 . 7 0 7
  • E 1 0 . 2 5

What is the ratio of the radii of the circular paths followed by the proton and by the alpha-particle?

  • A 1 0 . 7 0 7
  • B 1 0 . 3 5
  • C 1 0 . 5
  • D 1 1
  • E 1 0 . 2 5