D.3 Motion in electromagnetic fields

A straight wire of length 0.40 m carries a current of 3.0 A at right angles to a uniform magnetic field of flux density 0.25 T.

What is the force on the wire?

A straight wire in the plane of the page carries a current towards the bottom of the page. A uniform magnetic field is directed out of the page.

What is the direction of the magnetic force on the wire?

In a velocity selector the electric field strength is 4.5 × 10⁴ V m⁻¹ and the magnetic flux density is 0.030 T. The diagram shows the electric and magnetic forces on a negatively charged particle moving through the selector.

What speed must the particle have to pass through undeflected?

A positive ion enters a region of uniform field moving at right angles to the field lines. In one experiment the field is electric; in another it is magnetic.

Which row describes the path of the ion while it is in each field?

In a fusion reactor, an alpha particle moves at a speed of 1.2 × 10⁷ m s⁻¹ at an angle of 30° to a uniform magnetic field of flux density 5.0 T.

What is the magnitude of the magnetic force on the alpha particle?

A proton moves with constant speed through a uniform magnetic field. The angle θ between the velocity of the proton and the field can be varied from 0° to 180°.

Which graph shows how the magnitude F of the magnetic force on the proton varies with θ?

In a velocity selector, ions of speed v pass undeflected through perpendicular uniform electric and magnetic fields.

Which change allows ions of speed v/2 to pass undeflected instead?

Helium nuclei (He²⁺ ions, charge +2e, mass 6.64 × 10⁻²⁷ kg), initially at rest, are accelerated through a potential difference of 2.0 kV in a vacuum.

Which row gives the kinetic energy and the speed of each ion after the acceleration?

A straight wire in a robotic arm's actuator carries a constant current in a uniform magnetic field. When the wire makes an angle of 30° with the field, the magnetic force on it has magnitude F. The wire is rotated so that it is perpendicular to the field. The current and the length of wire in the field are unchanged. What is the new magnitude of the force?

Which combination of units is equivalent to the tesla?

A proton from the solar wind moves at right angles to a uniform region of a planet's magnetic field. No other forces act. Which statement is correct?

In an electrostatic crop sprayer, a negatively charged pesticide droplet passes between two large horizontal parallel plates. The upper plate is positive and the lower plate is negative. What is the direction of the electric force on the droplet?

A cosmic-ray proton travels exactly along a magnetic field line close to the Earth's magnetic pole. The field there is approximately uniform. Which statement describes the magnetic force on the proton?

In a dental X-ray tube, electrons start from rest and are accelerated through a potential difference of 30 kV. What kinetic energy does each electron gain?

A domestic extension cable contains two long, straight, parallel conductors close together. When an appliance is switched on, the conductors carry equal currents in opposite directions. What is the effect of the magnetic forces between the two conductors?

In an electron microscope, an electron moves through a region of uniform magnetic field. Its velocity is perpendicular to the field. Which quantity remains constant while the electron is in the field?

A cable carrying a steady current is placed inside the uniform magnetic field of an MRI scanner. For which orientation of the cable is the magnetic force on it zero?

An electron beam in a welding machine travels horizontally due east through a region where the magnetic field is uniform and directed vertically downwards. What is the direction of the magnetic force on the electrons as they enter the field?

In a solar-wind detector on a space probe, a proton and an alpha particle enter the same uniform magnetic field with the same speed, moving perpendicular to the field. An alpha particle has four times the mass and twice the charge of a proton. What is \(\frac{\text{radius of alpha particle's path}}{\text{radius of proton's path}}\)?

In an electric car, two long parallel cables run side by side. One carries current from the battery to the motor and the other carries the same current back to the battery. What is the magnetic force between the cables?

An ion-implantation machine used in microchip manufacture has a velocity selector. The selector has crossed uniform fields: electric field strength 4.0×10⁴ V m⁻¹ and magnetic flux density 0.20 T. Which ion speed passes through undeflected?

In a hospital X-ray tube, electrons are accelerated from rest through a potential difference of 30 kV. What is the kinetic energy gained by each electron?

An electron is projected in the same direction as a uniform electric field. Which statement describes the motion of the electron immediately after it is projected?

A straight wire carrying a current lies at 30° to a uniform magnetic field. The magnetic force on the wire is F.

The wire is turned until it is at right angles to the field. The current and the length of wire in the field do not change. What is the magnetic force on the wire now?

A proton and an alpha particle move at right angles to the same uniform magnetic field. The speed of the alpha particle is one quarter of the speed of the proton.

What is (magnetic force on the alpha particle)/(magnetic force on the proton)?

A particle of charge q and mass m moves with speed v at right angles to a uniform magnetic field of flux density B. It follows a circular path of radius r.

Which expression gives the charge-to-mass ratio q/m of the particle?

Two long parallel wires 0.10 m apart in a vacuum carry currents of 5.0 A and 3.0 A in the same direction.

Which row gives the magnitude of the force per unit length on each wire and describes the force?

A proton is released from rest in a uniform electric field in a vacuum.

Which graph shows the variation of the kinetic energy E_k of the proton with the distance s it has moved?

A proton and an electron travel at the same speed at right angles to the same uniform magnetic field.

What is (radius of the path of the proton)/(radius of the path of the electron)?

An electron moves from left to right across the page. A uniform magnetic field is directed into the page.

What is the direction of the magnetic force on the electron as it enters the field?

Two electrons P and Q move in circles at right angles to the same uniform magnetic field. The speed of Q is twice the speed of P.

Which row gives (radius of the path of Q)/(radius of the path of P) and (time for one revolution of Q)/(time for one revolution of P)?

Singly charged positive ions of different masses leave a velocity selector with the same speed. They then enter a uniform magnetic field at right angles to the field and move in circular paths of radius r.

Which graph shows the variation of r with the mass m of the ion?

An electron enters the uniform electric field between two parallel plates in a vacuum, moving parallel to the plates. x is the distance travelled by the electron parallel to the plates after entering the field and v_⊥ is the magnitude of the component of its velocity at right angles to the plates.

Which graph shows how v_⊥ varies with x while the electron is between the plates?

Positive ions pass undeflected through a region in which a uniform electric field of strength E and a uniform magnetic field of flux density B are perpendicular to each other and to the velocity of the ions.

The electric field is switched off. The ions now move in a circular path of radius r in the magnetic field.

What is the charge-to-mass ratio q/m of the ions?

A long horizontal wire X is fixed. A second long, thin wire Y, parallel to X and with a mass per unit length of 2.0 × 10⁻³ kg m⁻¹, is free to move vertically between smooth guides. Y is held at rest 4.0 mm directly above X by magnetic force alone. The same current I flows in both wires.

Which row gives I and the directions of the currents?

Two long parallel wires X and Y are 0.15 m apart in a vacuum. X carries a current of 2.0 A and Y a current of 8.0 A, both out of the page. A third long parallel wire Z, carrying a current, is placed on the line between X and Y. The resultant magnetic force on Z is zero.

What is the distance of Z from X?

In a simple model of the magnetron in a microwave oven, electrons move in circles at right angles to a uniform magnetic field, completing 2.45 × 10⁹ revolutions per second.

What is the magnetic flux density?

An electron of mass m and charge −e is released from rest at a plate. It is accelerated by the uniform electric field between this plate and a parallel plate a distance d away, which is at a potential difference V above it.

What is the time taken by the electron to cross the gap?

In a mass spectrometer used in an anti-doping laboratory, two types of ion, X and Y, enter the same uniform magnetic field with the same speed, perpendicular to the field. The radius of the path of X is twice the radius of the path of Y. What is \(\frac{(q/m)_X}{(q/m)_Y}\)?

Ions start from rest and are accelerated through a potential difference V. They then enter a uniform magnetic field at right angles and move in a circle of radius r. The accelerating potential difference is increased to 4V. What is the new radius?

A beam on a spacecraft instrument contains positive ions of many different masses, charges and speeds. The beam passes through a region of perpendicular uniform electric and magnetic fields. Some ions emerge undeflected. Which quantity must be the same for all of the undeflected ions?

In a research instrument, protons and alpha particles start from rest. Each is accelerated through the same potential difference and then enters the same uniform magnetic field at right angles to it. An alpha particle has twice the charge and four times the mass of a proton.
What is \(\frac{\text{radius of alpha particle path}}{\text{radius of proton path}}\)?

In a synchrotron light source, electrons travel around a curved section of the ring at constant speed under the action of a uniform magnetic field perpendicular to their velocity. Ignore radiation losses. Which quantity changes as an electron moves around the curved section?

A velocity selector in an ion-implantation machine has parallel plates 5.0 mm apart. A uniform magnetic field of 0.20 T acts perpendicular to the electric field. Ions travelling at 3.0 × 10⁵ m s⁻¹ pass through undeflected. What is the potential difference between the plates?

Protons for cancer therapy circulate in a uniform magnetic field perpendicular to their velocity. The speed of a proton is doubled, and its speed remains much less than the speed of light. What happens to the time it takes to complete one revolution?

An electron moves horizontally due east through a region where a uniform magnetic field points vertically upwards. What is the direction of the magnetic force on the electron at that instant?

Two parallel copper busbars in an electrical substation carry equal currents. The force per unit length between them is F. The current in each busbar is doubled and the separation of the busbars is halved. What is the new force per unit length?

A proton and an alpha particle from the solar wind enter the same uniform magnetic field with the same speed, both moving perpendicular to the field. The mass of the alpha particle is 4 times the mass of the proton and its charge is 2 times the charge of the proton. What is \(\frac{r_\alpha}{r_p}\), the ratio of the radius of the alpha particle's path to the radius of the proton's path?

An ion implanter used in microchip manufacture contains a velocity selector, in which a uniform electric field and a uniform magnetic field are perpendicular to each other and to the ion beam. Which ions pass through the selector undeflected?

In an old analogue oscilloscope, an electron enters the uniform field between two parallel deflecting plates, moving parallel to the plates. The arrangement is repeated with the potential difference between the plates doubled. The electron enters at the same speed as before. In both cases it leaves the region between the plates without striking either plate. What happens to the time the electron spends between the plates?

Three long, straight, parallel wires X, Y and Z lie in a plane, in that order, with adjacent wires a distance d apart. Each wire carries current I. The currents in X and Z are in the same direction and the current in Y is in the opposite direction. Two wires a distance d apart, each carrying current I, exert a force per unit length F₀ on each other. What is the resultant force per unit length on X?

A spacecraft ion thruster accelerates ions from rest through a potential difference V. Ion P is singly charged and ion Q is doubly charged. The two ions have the same mass. What is \(\frac{v_Q}{v_P}\), the ratio of the final speed of Q to the final speed of P?

A cyclotron used to produce medical isotopes accelerates protons in a uniform magnetic field. A proton's speed doubles while the field stays constant. What happens to the time taken for the proton to complete one circular orbit?

A charged dust grain in a planetary ring moves in a circle under the action of a uniform magnetic field alone. Which statement about the grain is correct?

Electrons from the Sun approach the Earth's polar regions, where the Earth's magnetic field is approximately uniform over a short distance. An electron enters this region with a velocity at 40° to the field lines. What is the shape of the electron's path while it is in this region?

A velocity selector in an ion-implantation machine uses perpendicular uniform electric and magnetic fields. It lets ions of speed v pass through undeflected. Which single change lets ions of speed 2v pass through undeflected instead?

A horizontal overhead cable runs east–west and carries a current flowing due east. Consider only the vertical component of the Earth's magnetic field at this location, which points downwards. What is the direction of the force on the cable due to this component?

A proton moving with speed v passes undeflected through a velocity selector. Three other particles enter the same selector, each moving with the same velocity v.
I. an electron
II. an alpha particle
III. a neutron
Which of these particles also pass through undeflected?

In a velocity selector, positive ions moving at speed E/B travel in a straight line. A positive ion enters the same selector, along the same line, with a speed greater than E/B. What is its initial motion in the selector?

An electron enters the uniform electric field between two parallel plates. It is moving parallel to the plates, midway between them. Gravity is negligible. Which statement is correct while the electron is between the plates?

In an industrial inkjet printer, charged ink droplets travel horizontally at constant speed between two parallel deflecting plates. A fixed potential difference is applied across the plates. The plate separation is doubled, while the potential difference and the plate length stay the same. What happens to the vertical deflection of a droplet as it leaves the plates?

In a test of an electrostatic spray-painting system, a charged paint droplet of mass 4.9×10⁻¹⁵ kg is held stationary between two horizontal plates. The plates are 1.2 cm apart, with a potential difference of 600 V between them. How many excess electrons are on the droplet?

A velocity selector in a research mass spectrometer currently selects ions of speed v. The operator wants it to select ions of speed 2v instead. Which change achieves this?

An electron enters the uniform electric field between two parallel plates, moving parallel to the plates. It leaves the plates with a deflection y.

The potential difference between the plates is doubled and the entry speed of the electron is doubled. The electron still leaves the plates. What is (new deflection)/y?

Two long parallel wires X and Y carry currents I and 3I respectively.

What is (force per unit length on Y)/(force per unit length on X)?

The diagram shows a positive ion moving in a circle in a uniform magnetic field directed into the page.

Which row describes the kinetic energy and the momentum of the ion as it moves round the circle?

A horizontal overhead cable runs from west to east and carries a direct current I towards the east. At this location the Earth's magnetic field has flux density B and is directed northwards at 60° below the horizontal. The diagram is a side view seen from the east, so the cable is seen end-on.

Which row gives the magnitude and the direction of the magnetic force on a length L of the cable?

A proton and an alpha particle, with the same kinetic energy, each enter the uniform electric field between the same pair of parallel plates, moving at right angles to the field. Both particles leave the region between the plates.

What is (deflection of the alpha particle)/(deflection of the proton) as they leave the plates?

A cloud chamber is placed in a uniform magnetic field directed into the page. A charged particle moves through the gas in the chamber at right angles to the field and leaves a track. The radius of curvature of the track decreases steadily along its length, so the track spirals inwards.

Which statement explains why the radius decreases?

A long straight wire in the plane of the page carries a current I to the right. A narrow beam of electrons in a vacuum travels to the right, parallel to the wire and a short distance d above it, in the plane of the page.

Which row gives the direction of the magnetic field of the wire at the beam and the direction of the magnetic force on the electrons?

A lightning stroke transfers negative charge from a cloud to the ground along a straight vertical channel. At one instant the current in the channel is 2.0 × 10⁴ A. At this place the Earth's magnetic field has a flux density of 5.0 × 10⁻⁵ T and is directed towards the north at 60° below the horizontal.

Which row gives the magnitude and the direction of the magnetic force on each metre of the channel at this instant?

An alpha particle and a proton each move in a circular path in the same uniform magnetic field. The alpha particle has four times the mass and twice the charge of the proton. Their speeds are different. What is \(\frac{\text{period of alpha particle}}{\text{period of proton}}\)?

An electron and a proton have the same kinetic energy. Each enters, separately, the same uniform electric field between parallel plates, moving at right angles to the field. Ignore gravity. Which statement compares their vertical deflections over the length of the plates?

In an electron-beam lithography machine, electrons enter the uniform electric field between two parallel deflecting plates. They enter moving parallel to the plates and do not strike them. On leaving the plates their vertical deflection is y.
The p.d. between the plates is doubled and the entry speed of the electrons is also doubled. The length and separation of the plates are unchanged. What is the new deflection?

In a vacuum, an electron is released from rest at the negative plate of a pair of parallel plates. It takes time t to reach the positive plate. The plate separation is then doubled, while the p.d. between the plates is kept the same. What is the new time taken to cross the gap?

In a cyclotron that produces isotopes for medical imaging, protons move in circular paths in a uniform magnetic field. A proton's speed is doubled, with the field unchanged. What happens to the radius of its path and to its period of revolution? Ignore relativistic effects.

In radiocarbon dating by mass spectrometry, singly charged ions of carbon-12 and carbon-14 are accelerated from rest through the same potential difference. They then enter the same uniform magnetic field perpendicular to their velocity. What is \(\frac{\text{radius of carbon-14 path}}{\text{radius of carbon-12 path}}\)?

A particle detector has a uniform magnetic field directed vertically upwards. A charged particle moves in a horizontal plane. Viewed from above, its track is a clockwise spiral of steadily decreasing radius. What can be deduced about the particle?

A proton and an alpha particle are each accelerated from rest through the same potential difference. Each then enters the same uniform electric field between parallel deflecting plates, moving initially parallel to the plates. What is the ratio \(\frac{\text{deflection of alpha particle}}{\text{deflection of proton}}\) at the end of the plates?

A student investigates the force on a current-carrying conductor. Two flat magnets are fixed to a steel yoke that rests on a top-pan balance of resolution 0.01 g. Seen from above, the north pole is on the student's left and the south pole on the student's right. A stiff copper rod is clamped horizontally so that it runs directly away from the student through the gap between the poles, without touching the magnets. The length of rod in the field is equal to the length of the pole faces, (5.0 ± 0.1) cm.

When there is a current I in the rod, the balance reading decreases by Δm. The uncertainty in each value of Δm is ±0.02 g.

A straight horizontal wire of length (40.0 ± 0.5) mm lies between the wide, flat pole pieces of an electromagnet, where the magnetic field is uniform and horizontal. The wire is fixed to a holder on a force sensor and can be turned in the horizontal plane so that it makes an angle θ with the field; the leads to the wire are twisted together outside the field.

The current in the wire is kept at (2.00 ± 0.01) A. The sensor measures the vertical force F on the wire with an uncertainty of ±0.2 mN.

A long solenoid has 600 turns wound uniformly over a length of 0.300 m and lies, with its axis horizontal, on a top-pan balance of resolution 0.001 g. A straight horizontal wire, held by a clamp that does not touch the solenoid, passes through two small holes across the middle of the solenoid, at right angles to its axis. The length of wire inside the solenoid is (40.0 ± 0.5) mm and the current in the wire is kept at (4.00 ± 0.02) A.

The current I_s in the solenoid is varied and the change Δm in the balance reading is recorded, with an uncertainty of ±0.002 g. Theory gives the flux density inside a long solenoid as B = μ₀nI_s, where n is the number of turns per unit length.

A horizontal copper rod of mass (4.80 ± 0.01) g hangs from two light vertical conducting wires, forming a swing. A pair of large coils produces a uniform vertical magnetic field of flux density B in the region of the rod. The length of the rod between the two suspension wires is (0.120 ± 0.001) m.

When there is a current I in the rod it swings sideways and comes to rest with the suspension wires at an angle φ to the vertical, measured with a protractor to ±1°. In equilibrium, tan φ = BIL/(mg), where m is the mass and L the length of the rod. The graph shows tan φ against I with error bars.

In an electron deflection tube, electrons accelerated through a potential difference V_a = (3.00 ± 0.05) kV enter the region between two horizontal parallel plates, travelling horizontally midway between them. The plates are (50.0 ± 0.5) mm apart and the upper plate is at a potential difference V_p = (1.20 ± 0.02) kV above the lower plate.

The path of the beam is shown on a fluorescent screen marked with a grid. A student reads the upward deflection y of the beam at horizontal distances x from the point where the beam enters the field. Each value of y has an uncertainty of ±1 mm. Theory predicts y = kx², where k = V_p/(4dV_a) and d is the plate separation.

A student uses a computer simulation of a mass spectrometer. Singly charged positive ions, initially at rest, are accelerated through a potential difference of 2.00 kV and then enter a uniform magnetic field of flux density 0.250 T at right angles to the field. Each ion travels through a semicircle and strikes a detector. The student measures the diameter D of each semicircle with the simulation's on-screen ruler, to ±1 mm.

The masses m of the ions, in unified atomic mass units (u), are taken from a published table of isotope masses. The student plots lg (D / mm) against lg (m / u); five points have been plotted.

Electrons leave a hot filament with negligible speed and are accelerated through a potential difference V by the uniform electric field between the filament and a grid. They then travel at constant speed v through a field-free tube. Two detectors (0.500 ± 0.001) m apart record the time of flight t of each pulse of electrons between them, with an uncertainty of ±0.5 ns.

A student uses the data to test whether ½m_ev² = eV and to determine e/m_e. Four of the points of the graph of v² against V have been plotted.

In a school oscilloscope tube, electrons accelerated from rest through a potential difference V_a pass between two parallel deflecting plates and then travel in a straight line to a fluorescent screen. The potential difference between the plates is kept at V_p = (24.0 ± 0.5) V. A student reads the vertical deflection D of the spot on the screen from the screen's grid to ±0.5 mm. V_a is set to ±1 %.

Theory predicts that D = kV_aⁿ, where n = −1 and k = V_plL/(2d). The plates have length l = (25.0 ± 0.5) mm and separation d = (6.0 ± 0.1) mm, and the screen is L = 0.180 m from the centre of the plates. The graph shows lg(D/mm) against lg(V_a/V) for four of the six measurements.