D.3 Motion in electromagnetic fields: IB Physics HL exam-style questions

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?

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 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 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 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?

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?

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?

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 (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?

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?

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?

A proton travelling at constant velocity enters, in turn, two separate regions of space.

In region X there is only a uniform electric field, directed at right angles to the velocity with which the proton enters. In region Y there is only a uniform magnetic field, directed parallel to the velocity with which the proton enters.

Which row describes the path of the proton in each region?

Two large parallel metal plates are held vertically, a few centimetres apart, in a vacuum. There is a uniform horizontal electric field between them. A small positively charged sphere is released from rest midway between the plates.

Which describes the path of the sphere before it strikes a plate?

Two long straight parallel wires P and Q carry steady currents. The magnetic force per unit length on Q is F.

The current in P is doubled, the current in Q is tripled and the separation of the wires is doubled.

What is the new magnetic force per unit length on Q?

A straight wire carries a current I in a uniform magnetic field. The length of wire in the field is 4.0 cm, and the wire makes an angle of 30° with the field lines. The graph shows how the magnetic force F on the wire varies with I.

What is the magnetic flux density?

A charged particle moves in a circle in a uniform magnetic field, at right angles to the field. No other forces act.

Which statements are correct?

I. The speed of the particle is constant.

II. The acceleration of the particle is constant.

III. The magnetic force does no work on the particle.

An alpha particle is released from rest in a uniform electric field in a vacuum. The graph shows how its kinetic energy E_k varies with the distance x it has moved.

What is the electric field strength?

Positive ions travelling at speed v pass undeflected through a region of uniform electric field E and uniform magnetic field B. The two fields are perpendicular to each other and to the velocity of the ions.

Which changes would still allow these ions to pass undeflected?

I. The electric field strength and the magnetic flux density are both doubled.

II. The speed of the ions and the electric field strength are both doubled.

III. The direction of the magnetic field is reversed and nothing else is changed.

A wire is bent into a semicircle of radius R and placed in a uniform magnetic field of flux density B directed into the page, as shown. There is a current I in the wire.

What is the magnitude of the resultant magnetic force on the semicircular part of the wire?

A boundary separates two regions of uniform magnetic field, both directed into the page. The flux density is B above the boundary and 2B below it. A positive particle at point P on the boundary moves at right angles to the boundary into the upper region, as shown. In the upper region its path has radius r.

The particle passes into the lower region and then returns to the upper region. What is the distance from P to the point where the particle returns to the upper region?

A particle of mass m and charge q moves at speed v at right angles to a uniform magnetic field and completes one quarter of a circle.

Which row gives the work done on the particle by the magnetic force and the magnitude of the change in momentum of the particle during this quarter circle?

An electron is between two parallel metal plates in a vacuum. The plates are 2.0 cm apart and the potential difference between them is 150 V.

What is the magnitude of the acceleration of the electron?

Ions of one type move in circular paths at right angles to a uniform magnetic field. The graph shows how the number of revolutions per second f of the ions varies with the magnetic flux density B.

What is the charge-to-mass ratio of the ions?

Electrons enter the uniform electric field between two parallel plates, moving parallel to the plates with speed v. They leave the plates moving at an angle θ to their original direction, without striking a plate.

The potential difference between the plates is halved and the speed with which the electrons enter is halved. The electrons still leave without striking a plate.

By what factor does tan θ change?

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 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.

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.

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.

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 time-of-flight mass spectrometer, singly charged positive ions are produced at rest and accelerated through a potential difference V by a uniform electric field in a very short gap. They then drift at constant speed along a field-free tube of length D = 1.200 m to a detector. An electronic timer measures the flight time t from the moment the ions are produced; the time spent in the accelerating gap is negligible.

The student records t for five types of ion of known mass m, measured in unified atomic mass units u. Theory predicts that t = D√(m/(2eV)), where m is the mass of the ion in kg.

Two long, straight, horizontal copper rods are connected in series so that each carries a current of 20.0 A, in opposite directions. The lower rod is clamped to a vertical micrometer stage, which sets the separation r between the centres of the rods. The upper rod, of length 0.300 m, hangs from a digital force sensor by an insulating thread; the sensor is set to zero with no current, so that it measures the magnetic force F on the upper rod. For each separation F is recorded. Each value of F has an uncertainty of ±0.05 mN.

The student suspects that F = krⁿ, where k and n are constants, and plots a graph of ln(F / mN) against ln(r / mm).

Two straight, parallel, horizontal metal rails are fixed 6.0 cm apart in a uniform vertical magnetic field of flux density B. A flat aluminium bar of length 10.0 cm and mass 8.0 g lies across the rails, at right angles to them. When a switch is closed there is a current I in the bar and the bar slides along the rails from rest. A light gate records the time t for the bar to slide 0.300 m. The student calculates the acceleration of the bar from a = 2s/t².

The current is kept constant during each run. Friction between the bar and the rails is constant. The magnetic fields of the currents in the rails can be ignored.

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 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.

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.

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.

A spacecraft is driven by an ion thruster. Singly charged xenon ions, each of mass 131 u, are accelerated from rest through a potential difference of 1.10 kV by the uniform electric field between two parallel grids. The ions leave the thruster as a beam; the beam current is 1.60 A. Electrons are injected into the beam as it leaves.