D.3 Motion in electromagnetic fields

In a cloud-chamber demonstration an alpha particle moves at 1.5 × 10⁷ m s⁻¹ through a uniform magnetic field of flux density 0.050 T. Its velocity makes an angle of 40° with the field lines.

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

An electron (mass m, charge −e) enters a region of uniform electric field of strength E. On entry its speed is v and it is moving in the same direction as the electric field lines.

What distance does the electron travel in the field before it comes momentarily to rest?

In a laboratory, a uniform magnetic field points vertically upwards. An electron moves horizontally due east through the field.
What is the direction of the magnetic force on the electron?

In a storage-ring experiment, a muon moves at constant speed in a circle in a uniform magnetic field. The muon's velocity is perpendicular to the field.
Which quantity remains constant?

A cosmic-ray proton passes through a laboratory magnet at a speed of 2.0 × 10⁶ m s⁻¹. Its velocity makes an angle of 30° with a uniform magnetic field of flux density 0.50 T.
What is the magnitude of the magnetic force on the proton? (e = 1.6 × 10⁻¹⁹ C)

A charged dust grain in space moves in a region containing only a uniform magnetic field. The grain does not move parallel to the field.
Which quantity remains constant?

Two long, straight, parallel cables on a pumping-station wall carry currents in the same direction. Which statement about the forces between the cables is correct?

A proton of charge \(q\) moves with speed \(v\) through a uniform magnetic field of flux density \(B\). Its velocity makes an angle of \(30^\circ\) to the field direction. What is the magnitude of the magnetic force on the proton?

A charged particle moves in a circular path of period \(T\) in a uniform magnetic field, with its velocity perpendicular to the field. The speed of the particle is doubled and the field is unchanged. What is the new period of the circular motion?

An electron accelerated from rest through a potential difference V enters a uniform magnetic field at right angles and follows a circular path of radius r. The potential difference is increased to 4V.

What is the new radius?

In a velocity selector the uniform electric field is directed towards the bottom of the page and the uniform magnetic field is directed into the page. Positive ions moving to the right at speed v pass through undeflected.

Ions of the same type, with the same speed v, are now sent into the selector from the opposite end, so that they move to the left. What happens to these ions as they enter the fields?

A proton and an alpha particle move in circular paths in the same uniform magnetic field.

What is the ratio (period of the alpha particle) / (period of the proton)?

A horizontal metal rod of mass m and length L hangs from two light, flexible, conducting threads in a uniform vertical magnetic field of flux density B. When there is a current I in the rod, the rod moves sideways and comes to rest with the threads at an angle θ to the vertical.

Which expression gives I?

An electron with kinetic energy E_k enters the uniform electric field between two parallel plates. It enters midway between the plates, moving parallel to them. The potential difference between the plates is V. The electron is deflected and strikes the positive plate.

What is the kinetic energy of the electron when it strikes the plate?

Two long parallel wires X and Y lie on a smooth horizontal surface and carry currents in the same direction. The current in Y is three times the current in X, and the mass per unit length of Y is half that of X. The wires are released from rest.

What is (initial acceleration of X)/(initial acceleration of Y)?

A charged particle moves in a circle in a uniform magnetic field directed out of the page. At the instant shown it is moving towards the top of the page, and the force on it is towards the centre O.

Which row gives the sign of the charge and the effect on the radius of the path if the speed of the particle is doubled?

Positive ions pass undeflected through a velocity selector in which the electric field strength is 3.0 × 10⁴ V m⁻¹ and the magnetic flux density is 0.060 T. They then enter a region where the only field is a uniform magnetic field of flux density 0.20 T, perpendicular to their velocity. They travel through a semicircle and strike a detector 5.2 cm from the point where they entered this region.

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

Two long, straight, parallel busbars in an electrical substation are 0.25 m apart. During a fault each carries a current of 2.0 kA, in opposite directions. Each busbar is held in place by supports placed every 1.5 m along its length.

What is the magnitude of the magnetic force on a 1.5 m section of one busbar?

A long straight wire in the plane of the page carries a current towards the top of the page. An electron to the right of the wire moves parallel to the wire, also towards the top of the page, as shown.

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

A proton is released from rest next to the positive plate of a pair of parallel plates in a vacuum. The potential difference between the plates is V and their separation is d. The proton reaches the negative plate after a time t.

The separation of the plates is doubled and the potential difference is unchanged. What is the time now taken by the proton to cross from one plate to the other?

A proton enters a region of uniform magnetic field, moving at right angles to the field and to the boundary of the region, as shown. The region has parallel boundaries a distance w apart. Inside the region the proton moves along a circular arc of radius 2w.

What is the angle θ through which the proton has been deflected when it leaves the region?

Two long parallel wires carry currents I and 3I in the same direction. Which statement describes the magnetic forces between the wires?

Two parallel power cables in an electrical substation carry currents and exert a force per unit length F on each other. Both currents are doubled. The separation of the cables is also doubled. What is the new force per unit length?

A particle detector contains a uniform magnetic field directed out of the plane of the page. A charged particle leaves a track that spirals clockwise and inwards, so its radius gets steadily smaller.
What is the sign of the particle's charge, and what is happening to its kinetic energy?

In a beamline for a radiotherapy research facility, protons and alpha particles (He²⁺) start from rest. Both are accelerated through the same potential difference. They then enter the same uniform magnetic field at right angles to it.
What is \(\frac{r_\alpha}{r_p}\), the ratio of the radius of the alpha particle's path to that of the proton's path? Take the mass of an alpha particle as 4 times the proton mass.

A straight wire in a test rig makes an angle of 30° with a uniform magnetic field. The magnetic force on the wire is F.
The wire is rotated until it is perpendicular to the field, and at the same time the current is halved.
What is the new magnetic force on the wire?

Two long, straight, parallel cables in a data centre are a distance d apart. One carries current I and the other carries current 3I, in the same direction. The magnetic force per unit length on the cable carrying I is F.
What is the force per unit length on the cable carrying 3I, and what is its nature?

A velocity selector uses perpendicular uniform electric and magnetic fields. Singly charged ions moving at speed v pass through it undeflected.
A doubly charged ion enters the same selector at speed 2v. What happens to this ion?

Two long, parallel power cables carry currents of 2I and I in opposite directions. Their separation is r, and the force per unit length on each cable has magnitude F.
Both currents are doubled and the separation is increased to 3r. What is the new force per unit length?

An electron enters the uniform field between two parallel deflecting plates. Its initial velocity is parallel to the plates. The potential difference between the plates is then doubled, and the electron's entry speed is unchanged.
What happens to the time the electron spends between the plates and to its sideways deflection when it leaves the plates?

In a cyclotron used to produce medical isotopes, a proton moves in a circle perpendicular to a uniform magnetic field. The proton's speed is doubled while it stays in the same field.
What happens to the radius and to the period of its circular motion?

Two long parallel wires carry currents \(I_1\) and \(I_2\) at separation \(d\), with force per unit length \(F\) on each wire. Both currents are doubled and the separation is also doubled. What is the new force per unit length?

A velocity selector has an electric field pointing towards the top of the page and a magnetic field pointing out of the page. A positive ion moves to the right through the selector at a speed greater than \(E/B\). Towards which direction is the ion deflected?

Electrons are accelerated from rest through a potential difference \(V\) in the electron gun of a retro arcade monitor. They then move in a circle of radius \(r\) in a uniform magnetic field \(B\) perpendicular to their velocity. Which expression gives the specific charge \(e/m\) of the electron?

A straight section of a pump cable of length 0.50 m carries a current of 3.0 A. It lies in a uniform magnetic field of flux density 0.20 T, at \(30^\circ\) to the field direction. What is the magnitude of the magnetic force on the cable?

Three long straight parallel wires X, Y and Z are perpendicular to the page and lie in the same plane. X and Y are a distance d apart, and Y and Z are a distance 2d apart. X and Y each carry a current I into the page; Z carries a current 2I out of the page, as shown.

What is the magnitude of the resultant magnetic force per unit length on Y?

A rectangular loop of wire PQRS carries a current and lies in the plane of the page in a uniform magnetic field. The field is also in the plane of the page and is parallel to sides PQ and SR, as shown.

Which statement describes the effect of the magnetic forces on the loop?

A proton and an alpha particle are each accelerated from rest through the same potential difference. Each then enters, midway between them and parallel to them, the uniform electric field between the same pair of horizontal deflecting plates.

How do the vertical deflection on leaving the plates and the time spent between the plates for the alpha particle compare with those for the proton?

A proton (mass m, charge +e), a deuteron (mass 2m, charge +e) and an alpha particle (mass 4m, charge +2e) all have the same kinetic energy. Each moves in a circle in the same uniform magnetic field, perpendicular to its velocity.

Which gives the correct relationship between the radii r_p, r_d and r_α of their paths?

In a mass spectrometer used to check the purity of a pharmaceutical sample, positive ions first pass undeflected through a velocity selector. The selector has an electric field of strength E and a magnetic field of strength B₁, perpendicular to each other.
The ions then enter a region containing only a uniform magnetic field of strength B₂. They complete a semicircle and strike a detector a distance D from the entry slit, where D is the diameter of the semicircle.
What is the specific charge q/m of the ions?

A long straight wire carries a steady conventional current. An electron moves parallel to the wire, a short distance from it, in the same direction as the conventional current.
What is the direction of the magnetic force on the electron?

In a radiotherapy research facility, a proton and an alpha particle have the same kinetic energy. Each moves perpendicular to the same uniform magnetic field. The alpha particle has 4 times the mass and 2 times the charge of the proton.
What is the ratio (radius of alpha particle's path)/(radius of proton's path)?

In a fine-beam tube, electrons are accelerated from rest through a potential difference V. They then move in a circle of radius r in a uniform magnetic field of flux density B.
The accelerating potential difference is increased to 4V. By what factor must B be multiplied to keep the radius equal to r?

In a mass spectrometer used to analyse a drug metabolite, ion P has mass \(m\) and charge \(+q\); ion Q has mass \(4m\) and charge \(+2q\). Both are accelerated from rest through the same potential difference and then enter the same uniform magnetic field at right angles to it. What is the ratio \(\frac{\text{radius of path of Q}}{\text{radius of path of P}}\)?

In a fine-beam tube, electrons are accelerated from rest through a potential difference V and then enter a uniform magnetic field of flux density B = 1.00 × 10⁻³ T at right angles to their velocity. They move in a circle whose radius r is read from a scale behind the beam. Theory predicts that r² = (2m/(eB²))V, where e/m is the specific charge of the electron.

B is known to ±2 %, and the gradient of the graph has an uncertainty of ±3 %. The graph shows four of the five data points.

Two straight aluminium rods of diameter 2.0 mm are held horizontal, one vertically above the other, with a distance r = 10.0 mm between their centres. They overlap for a length of 0.400 m and are connected in series so that they carry the same current I in opposite directions. The lower rod rests on insulating supports on a top-pan balance, which is set to zero when there is no current.

For each current the increase in the balance reading Δm is recorded, and the magnetic force on the lower rod is found from F = Δm × g. The graph shows four of the five points.

A U-shaped permanent magnet rests on a digital top-pan balance, with its uniform horizontal field between the poles. A straight horizontal copper wire is held between the poles, perpendicular to the field, by a clamp that does not touch the magnet or the balance. The length of wire in the field is 5.00 cm ± 0.10 cm. The balance was set to zero with the magnet on the pan, but the student did not re-zero it before taking readings.

The student records the balance reading m for six values of the current I in the wire. The readings increase as the current increases. Each reading has an uncertainty of ±0.01 g; the uncertainty in I is negligible. The data are shown in the table and in the graph.

In an electron deflection tube, electrons accelerated from rest through a potential difference V_a enter, midway between them, the uniform electric field between two horizontal deflecting plates. The plates are 4.0 cm long and 1.0 cm apart and the potential difference between them is fixed at 40.0 V. After leaving the plates the electrons travel a further 18.0 cm to a fluorescent screen. The vertical deflection y of the spot on the screen is read from a scale ruled on the screen; each reading has an uncertainty of ±1 mm.

Theory predicts that y = kV_d/V_a, where V_d is the potential difference between the plates and k depends only on the dimensions of the tube. For this tube the dimensions give k = 0.40 m. The graph shows five of the six data points.

Electrons are accelerated from rest through a potential difference V_a. They then pass between two horizontal parallel plates 5.0 mm apart, where a uniform magnetic field of flux density 1.20 mT acts perpendicular to both the electric field between the plates and the velocity of the electrons. For each value of V_a the potential difference V_p between the plates is adjusted until the beam passes through undeflected.

The flux density is known to ±2 % and the plate separation to ±3 %. The graph shows V_p² against V_a.

In a continuous ink-jet printer, ink droplets of mass 1.5 × 10⁻¹¹ kg are each given a negative charge of magnitude 2.0 × 10⁻¹³ C. Each droplet travels horizontally at 18 m s⁻¹ and enters, midway between them, the region between two horizontal deflecting plates. The plates are 1.2 cm long and 3.0 mm apart, and the potential difference between them is 900 V. The paper is 1.5 cm beyond the end of the plates. Assume the field between the plates is uniform and zero outside them.

An electrodynamic shaker is used to vibrate small components for testing. A circular coil of 80 turns and radius 1.8 cm is fixed to a light table. The coil sits in the narrow circular gap of a permanent magnet, where the magnetic field is radial, has a flux density of 0.65 T and is everywhere perpendicular to the wire of the coil, as shown in the top view.

The table and coil have a total mass of 0.12 kg. They are supported by flexible mounts that behave as a spring of spring constant 3.0 × 10³ N m⁻¹.

An engineering student builds a model rail launcher. A metal bar of mass 0.050 kg rests across two horizontal rails that are 0.12 m apart. A current of 25 A flows along the rails and through the bar. A uniform vertical magnetic field of flux density 0.40 T acts perpendicular to the bar. Ignore friction and any changes in the current.

A singly charged ion of a drug molecule has mass \(5.4\times10^{-25}\) kg. It moves at \(2.5\times10^{4}\) m s⁻¹ at right angles to a uniform magnetic field of flux density 0.40 T. Use \(e=1.60\times10^{-19}\) C.

In a fine-beam tube, electrons are accelerated from rest through a potential difference of 3.00 kV. They then move perpendicular to a uniform magnetic field of flux density 1.95 mT, and the radius of their circular path is measured as 9.5 cm.

A cyclotron accelerates protons using a uniform magnetic field of flux density 1.50 T. The protons cross the gap between the two D-shaped electrodes twice in each revolution, and at each crossing they are accelerated through a potential difference of 50.0 kV.

Two horizontal parallel plates, 4.0 cm long and 1.2 cm apart, have a potential difference of 240 V between them; the upper plate is positive. A uniform magnetic field, perpendicular to the electric field, can also be applied in the region between the plates. A narrow beam of electrons, accelerated from rest through 2.00 kV, enters the region midway between the plates, moving horizontally to the right, as shown.

A hospital cyclotron produces protons for making medical isotopes used in PET scanning. Inside the cyclotron, protons move in semicircular paths in a uniform magnetic field of strength 1.4 T. Each time a proton crosses the gap between the two halves, it is accelerated by an alternating potential difference. Relativistic effects may be ignored.
Mass of proton = 1.67×10⁻²⁷ kg; e = 1.60×10⁻¹⁹ C.

In a semiconductor ion implanter, singly charged positive ions are accelerated from rest through a potential difference of 20.0 kV. They then enter an analysing magnet. Only ions that follow a path of radius 0.400 m reach the exit slit. The magnetic field strength B is adjusted to select each ion. The table shows the field needed for three ions.

In an electric-vehicle battery pack, two long, straight, parallel copper busbars are 12 mm apart. Each is 0.80 m long. In normal operation, each busbar carries 300 A, and the currents are in opposite directions.
μ₀ = 4π×10⁻⁷ T m A⁻¹.

A solar-wind proton enters a region of the Earth's outer magnetosphere where the magnetic field strength is 30 nT. The proton moves at 4.0×10⁵ m s⁻¹, perpendicular to the field.
Mass of proton = 1.67×10⁻²⁷ kg; e = 1.60×10⁻¹⁹ C.

In a data centre, power is distributed by two long, straight, parallel copper busbars. The busbars are 4.0 cm apart. In normal operation the supply bar and the return bar each carry 1200 A, in opposite directions. Treat each busbar as a long thin wire.
μ₀ = 4π × 10⁻⁷ T m A⁻¹

A student measures the specific charge of the electron using a fine-beam tube. Electrons are accelerated from rest through a potential difference V. They then move in a circle of radius r in a uniform magnetic field of flux density 0.80 mT, perpendicular to their velocity.

V / V : 150, 200, 250, 300
r / cm : 5.2, 6.0, 6.6, 7.3

A solar-wind proton travels at 4.0 × 10⁵ m s⁻¹ into a region of Earth's magnetosphere. There the magnetic field is uniform, with flux density 30 nT. At first the proton's velocity is perpendicular to the field.
e = 1.60 × 10⁻¹⁹ C; m_p = 1.67 × 10⁻²⁷ kg

In the injector of a proton-therapy accelerator, protons start from rest and are accelerated across a 4.0 cm gap. The gap lies between two parallel electrodes with a potential difference of 50 kV between them. Assume the field in the gap is uniform and non-relativistic mechanics applies.
e = 1.60 × 10⁻¹⁹ C; m_p = 1.67 × 10⁻²⁷ kg

Two long straight cables carry currents in the same direction. They are parallel and 0.080 m apart. Cable P carries a current of 60 A. The force per unit length on cable Q is measured as \(2.4\times10^{-3}\) N m⁻¹. Use \(\mu_0=4\pi\times10^{-7}\) T m A⁻¹.

In a demonstration of the specific charge of the electron, electrons are accelerated from rest through a potential difference of 1.20 kV and enter a region of uniform magnetic field of flux density 1.80 mT perpendicular to their velocity. Their path is a circle of radius 6.49 cm.

Two singly charged ions of a radioactively labelled drug fragment are produced in a mass spectrometer. The lighter ion has mass \(1.99\times10^{-26}\) kg and the heavier ion has mass \(2.32\times10^{-26}\) kg. Each is accelerated from rest through a potential difference of 3.00 kV and then enters a uniform magnetic field of flux density 0.200 T at right angles to the field, and travels in a semicircle. Use \(e=1.60\times10^{-19}\) C.
Use unrounded values in later parts.

An electromagnetic flow meter measures the speed of water in a pipe of internal diameter 0.12 m. The pipe is non-magnetic and non-conducting. A uniform magnetic field of flux density 0.080 T is applied across the pipe, perpendicular to the flow. Electrodes on opposite sides of the pipe, at the ends of a diameter that is perpendicular to both the flow and the field, measure a potential difference caused by dissolved ions.

Singly ionised magnesium ions are accelerated from rest through a potential difference of 2.00 kV. They then enter a region of uniform magnetic field of flux density 0.400 T, directed perpendicular to their velocity, travel through a semicircle and strike a flat detector plate that lies along the line through the point of entry. The beam contains the isotopes ²⁴Mg and ²⁶Mg, with ion masses of 24 u and 26 u.

In a demonstration of magnetic levitation, a long straight wire X is fixed horizontally on a bench and carries a current of 50 A into the page. A light, straight aluminium wire Y, parallel to X, carries a current of 5.0 A and is free to move vertically between two glass guides, as shown. Y has a mass per unit length of 5.3 × 10⁻⁴ kg m⁻¹ and floats at rest at a distance r vertically above X.