D.2 Electric and magnetic fields

Three identical small conducting spheres X, Y and Z are mounted on insulating handles. At first X has a charge of +8.0 nC, Y has a charge of −2.0 nC and Z is uncharged.

X is touched against Y and they are then separated. Y is next touched against Z and they are separated.

What is the final charge on Z?

A long straight vertical wire passes through a hole in a horizontal card. There is a steady current upwards in the wire. A small plotting compass is placed on the card due east of the wire. The Earth's magnetic field can be ignored.

In which direction does the north pole of the compass needle point?

An alpha particle (charge +2e) moves in an electrostatic analyser from a point where the electric potential is +1200 V to a point where the electric potential is −300 V. Only the electric force acts on it.

What is the increase in the kinetic energy of the alpha particle?

At point P, a distance r from an isolated point charge Q, the electric field strength is E and the electric potential is V.

The charge is replaced by a point charge 2Q and P is moved to a distance r/2 from it. Which row gives the new electric field strength and the new electric potential at P?

An uncharged metal sphere stands on an insulating base. A negatively charged plastic rod is held close to the sphere without touching it. With the rod still in place, the sphere is briefly connected to earth and then disconnected. The rod is then taken away. What is the final charge on the sphere?

A hollow metal sphere of radius R carries charge +Q. Which row gives the electric field strength E and the electric potential V at a point a distance R/2 from the centre?

Two small charged dust grains in a planetary ring exert an electrostatic force of magnitude F on each other. The charge on one grain doubles and the separation of the grains becomes three times larger. What is the new magnitude of the force?

Which unit is equivalent to the unit of electric field strength, N C⁻¹?

In a tumble dryer a nylon sock and a cotton shirt rub together. The two items are initially uncharged and together form an isolated system. Afterwards the sock has a charge of +5.0 nC. Which row gives the charge on the shirt and the particles transferred?

Which statement about magnetic field lines is correct?

Two small ice crystals on a mountain ridge carry charges and repel each other with a force F when their separation is r. One charge is then doubled and the separation is increased to 3r. What is the new force between them?

The magnetic field at a distance r from a long straight current-carrying wire has magnitude B. What is the magnitude of the field at a distance 3r from the same wire when the current is halved?

In a data centre, a neutral polyester cloth is rubbed on a neutral plastic casing. Afterwards the casing carries a charge of −4.8 nC. Which row correctly describes the cloth and the charge transfer?

Two small charged grains of dust lie a distance r apart and exert an electrostatic force of magnitude F on each other. The charge on one grain is doubled, the other charge is unchanged, and the separation is increased to 3r. What is the new magnitude of the force?

Two parallel metal plates carry equal and opposite charges. In the region between the plates, away from the edges, the electric field lines are drawn as equally spaced parallel lines. What does this show about the field in that region?

Two point charges, each +Q, are fixed at two vertices of an equilateral triangle of side a. Point P is the third vertex of the triangle.

Which row gives the magnitude of the electric field strength E and the electric potential V_e at P?

In an experiment of Millikan's type, the charges on four oil drops are measured as 4.8 × 10⁻¹⁹ C, 8.0 × 10⁻¹⁹ C, 11.2 × 10⁻¹⁹ C and 14.4 × 10⁻¹⁹ C.

What is the largest value of the basic unit of charge that is consistent with these data?

Two small charged spheres in air exert an electric force F on each other. The permittivity of air may be taken as ε₀. The spheres are then immersed in an oil of permittivity 3.0ε₀ and their separation is halved.

What is the new electric force between the spheres?

The diagram shows the cross-sections of five plane equipotential surfaces in a region of electric field, with their potentials and a scale. Point P is midway between the 60 V and 40 V surfaces.

What is the magnitude of the electric field strength at P?

An uncharged aluminium sphere stands on an insulating base. A negatively charged PVC rod is held close to the sphere but never touches it. With the rod still in place, the sphere is briefly connected to earth by a wire. The wire is then removed, and finally the rod is taken away.

I. While the sphere is connected to earth, electrons flow from the sphere to earth.

II. After the rod is taken away, the sphere is positively charged.

III. If the rod had been taken away before the earth wire was removed, the sphere would also have been left positively charged.

Which statements are correct?

Two small beads P and R are fixed 0.200 m apart on a horizontal insulating thread in air. P has a charge of +3.0 nC and R has a charge of −5.0 nC. A third bead S with a charge of −2.0 nC is placed on the line between them, 0.080 m from P.

What is the magnitude of the resultant electric force on S?

In an experiment of Millikan's type, a charged oil drop is held stationary between two horizontal parallel plates. The potential difference between the plates is then tripled and, at the same moment, the separation of the plates is halved. Air resistance is negligible at the instant of the change.

What is the magnitude of the acceleration of the drop immediately after the change?

The diagram shows the electric field lines around two point charges X and Y. All the field lines that start or end on the charges are drawn.

Which row gives the sign of the charge on Y and the ratio (magnitude of charge on X)/(magnitude of charge on Y)?

One cell of an electrostatic air cleaner consists of two parallel metal plates with a potential difference between them. The plates are long compared with their separation. Point P is midway between the plates near their centre. Point Q is also near the centre, a quarter of the way from the negative plate to the positive plate. Point R is level with P but just beyond the edge of the plates, as shown.

An electron is placed in turn at P, Q and R. Which describes the magnitudes F_P, F_Q and F_R of the electric force on it?

A long air-core solenoid has ends P and Q. When the solenoid is viewed along its axis from end P, the current in the turns is clockwise.

Which row gives the magnetic polarity of end P and the direction of the magnetic field lines inside the solenoid?

Four point charges are fixed at the corners of a square of side a. The two upper corners each carry +q and the two lower corners each carry −q. M is the midpoint of the upper side of the square.

What is the electric potential at M?

Three point charges, +q, +q and −q, are fixed at the corners of an equilateral triangle of side a.

What is the electric potential energy of the system?

The diagram shows the magnetic field lines in a plane perpendicular to two long, straight, parallel wires X and Y. The wires carry currents of equal magnitude.

Which row gives the directions of the currents in X and in Y?

In a Millikan-type experiment, a charged oil drop is held stationary between horizontal parallel plates when the potential difference across them is V. The plate separation is halved and the drop's charge does not change. What potential difference is now needed to hold the drop stationary?

Three point charges +q, +q and −q are fixed at the corners of an equilateral triangle of side a. What is the electric potential energy of this system?

In the electrode region of a medical defibrillator test rig, the field is uniform. Two points P and Q lie on the same field line, 4.0 cm apart. The potential at P is 120 V and at Q is 40 V. What are the magnitude and direction of the electric field?

An electron starts from rest at a point where the electric potential is −200 V. It moves, under the electric force only, to a point where the potential is +300 V. What kinetic energy does the electron gain?

Beneath a thundercloud, the electric potential increases with height, rising by 1.5×10⁶ V for every 100 m gained. The field is approximately uniform. Which row gives the magnitude and direction of the electric field strength?

The dome of a Van de Graaff generator is an isolated conducting sphere of radius R carrying charge Q. Point P is inside the dome, a distance R/2 from its centre. Which row gives the electric field strength and the electric potential at P?

An engineer maps the equipotential surfaces above a charged circuit board. The potential difference between adjacent equipotential surfaces is always the same. Near point P the surfaces are 2.0 mm apart, and near point Q they are 6.0 mm apart. What is \(\frac{E_P}{E_Q}\), the ratio of the electric field strength at P to that at Q?

In a test of an electrostatic crop sprayer, a charged droplet is held stationary between two horizontal parallel plates. The plates are a distance d apart and have a constant potential difference across them. The plate separation is then increased to 2d, with the potential difference unchanged. What is the motion of the droplet immediately afterwards?

In a spacecraft ion thruster, a singly charged xenon ion moves from an equipotential surface at +250 V to one at −150 V. No other forces act on it. What is the increase in its kinetic energy?

The dome of an electrostatic generator is a conducting sphere of radius 0.20 m charged to a potential of 300 kV. What is the electric potential at a point inside the dome, 0.10 m from its centre?

Two isolated conducting spheres, of radii r and 2r, are far apart and joined by a long thin conducting wire. Charge is placed on the system. What is the ratio \(\frac{\text{field strength at surface of small sphere}}{\text{field strength at surface of large sphere}}\)?

Air breaks down and conducts when the electric field strength exceeds about 3.0×10⁶ V m⁻¹. What is the approximate maximum potential that a conducting sphere of radius 0.15 m can reach in air?

In the retarding-grid stage of an electron energy analyser, an electron moves from a point where the electric potential is +300 V to a point where the electric potential is −200 V. What is the change in the electric potential energy of the electron?

A cable for an electric-vehicle charger contains two long parallel conductors a distance d apart. Each carries a current I, in opposite directions. What is the magnitude of the magnetic field at the point midway between the conductors?

A fixed sphere carries a charge of +5.0 μC. How much work must be done to bring a small sphere carrying +2.0 μC from infinity to a point 0.30 m from the centre of the fixed sphere? (Coulomb constant k ≈ 9×10⁹ N m² C⁻²)

Three identical metal spheres, P, Q and R, sit on insulating stands. Initially P has +12 nC, Q has −4 nC and R is uncharged. P is touched to Q and separated. Then Q is touched to R and separated. What are the final charges?

A point charge +Q and a point charge −Q are fixed a short distance apart, forming a model of a polar molecule. Which describes the electric potential V and the electric field strength E at the midpoint between them?

Two long straight parallel wires in a telecoms cable duct are 0.10 m apart and carry currents of 3.0 A in opposite directions. The permeability of free space is μ₀ = 4π × 10⁻⁷ N A⁻². What is the magnitude of the resultant magnetic field at the point midway between the wires?

Two fixed point charges of equal magnitude, +Q and −Q, are a distance d apart. Which statement is correct for the point midway between them?

Two fixed point charges +q are a distance a apart. How much work must be done by an external agent to bring a third charge +q from infinity to the third vertex of the equilateral triangle of side a?

Two point charges, +4Q and +Q, are fixed 3.0 m apart in a telecoms test range. At what distance from the +Q charge, on the line between them, is the resultant electric field zero?

Two small fixed spheres carry charges +3.0 nC and +1.0 nC. Their centres are 0.40 m apart in air. What is the electric field strength at the midpoint of the line joining them, and in which direction does the field point? (Use k = 8.99×10⁹ N m² C⁻².)

An isolated conducting sphere of radius 0.20 m carries a positive charge. The electric potential at its surface is +900 V, with zero potential at infinity. What is the electric potential at a point 0.60 m from the centre of the sphere?

Near the ground, the atmosphere has a uniform electric field of strength 130 V m⁻¹ directed vertically downwards. A cyclist's head is 1.8 m above the ground. Which statement about the potential difference between the cyclist's head and the ground is correct?

The graph shows how the electric potential V varies with distance x along a straight line in a region of electric field. An electron at x = 0 is to move along the line to x = 6.0 cm. Only the electric force acts on the electron.

What is the minimum kinetic energy that the electron must have at x = 0 to reach x = 6.0 cm?

A hollow conducting sphere of radius R carries a positive charge Q. Point X is inside the sphere, a distance R/2 from its centre. Point Y is outside the sphere, a distance 2R from its centre.

What is the work done by the electric field on a small positive charge q as it moves from X to Y?

Two long straight parallel wires P and Q are perpendicular to the page and a distance d apart. P carries a current 2I out of the page and Q carries a current I into the page. The magnetic field strength at a distance r from a long straight wire carrying current I is proportional to I/r.

Points J, K, L and M lie on the line through the wires, as shown: K is midway between the wires, J is between the wires a distance d/3 from Q, L is a distance d beyond Q and M is a distance d beyond P.

At which point is the resultant magnetic field zero?

A point charge +4Q is fixed at x = 0 and a point charge −Q is fixed at x = d. At which position on the x-axis is the resultant electric field strength zero?

A model shows why sharp points on a lightning conductor discharge first. Two isolated metal spheres of radii R and 2R are far apart and joined by a thin conducting wire. A total charge is shared between them. What is the ratio (field strength at the surface of the small sphere) / (field strength at the surface of the large sphere)?

The dome of a science-museum Van de Graaff generator can be treated as a point charge at its centre for points outside it. At a point P outside the dome, the electric potential is 600 V and the electric field strength is 200 V m⁻¹. How far is P from the centre of the dome?

Two identical positive point charges +Q are fixed a distance d apart. Which correctly describes the electric field strength E and the electric potential V at the midpoint between them? (V is zero at infinity.)

In a model of an ionised molecule, a point charge +4q is fixed a distance d from a point charge −q. On the straight line between the two charges, at what distance from the +4q charge is the electric potential zero?

In a medical isotope facility, a proton and an alpha particle are each accelerated from rest through the same potential difference in a uniform electric field. What is the ratio \(\frac{\text{speed of proton}}{\text{speed of alpha particle}}\)?

A point charge +Q is fixed at a distance d from a point charge −2Q. At which point on the straight line between the two charges is the electric potential zero?

A long vertical wire carries a steady current of 10 A upwards. The horizontal component of the Earth's magnetic field is 2.0×10⁻⁵ T, directed north. Ignore the vertical component. At which point in a horizontal plane is the resultant horizontal magnetic field zero?

In fair weather, the electric field near the Earth's surface is about 130 N C⁻¹, directed vertically downwards. Which statement is correct?

A particle of mass m and charge +q is released from rest at a distance r from a fixed point charge +Q. What is its speed when its distance from +Q has increased to 2r? (Ignore gravity.)

A charge of +2q is fixed at x = 0 and a charge of −q is fixed at x = 3.0 m. At which positions on the x-axis is the electric potential zero (taking V = 0 at infinity)?

Two identical small conducting spheres carry charges +6.0 nC and −2.0 nC. Their centres are a fixed distance apart, and they attract each other with a force of magnitude F. The spheres are touched together and returned to their original separation. How does the force between them now compare with F?

In a Millikan-type experiment, oil drops of density 875 kg m⁻³ are viewed between two horizontal metal plates 5.00 mm apart. With no potential difference between the plates, drop A falls at constant (terminal) speed. A student times it with a hand-held stopwatch over 1.00 mm of the eyepiece scale and records 8.35 s.

A potential difference V is then applied and adjusted until drop A is stationary: V = 457 V. The viscosity of air is 1.82 × 10⁻⁵ Pa s, and the buoyancy force of the air on a drop may be neglected. The charges on four other drops, B to E, were found in the same way.

A hollow aluminium sphere of radius 0.060 m stands on an insulating base and carries a charge Q. A student measures the electric potential V at distances r from the centre of the sphere using a flame probe connected to an electrostatic voltmeter. Before taking the readings, the student touches the probe against an earthed metal plate: the voltmeter then reads −45 V.

The student's hypothesis is that V is inversely proportional to r, as it would be for a point charge at the centre of the sphere. The uncorrected voltmeter readings are shown. k = 8.99 × 10⁹ N m² C⁻².

Two long, straight, parallel copper strips are painted on a sheet of conducting paper 8.0 cm apart and connected to a 6.00 V supply; the negative strip is at 0 V. A probe connected to a voltmeter is used to measure the potential V on the paper at distances x from the negative strip, along a line halfway between the ends of the strips.

The graph shows the data with the line of best fit, extended to both strips.

A flat circular coil of N turns is sealed inside a plastic housing, so its radius R cannot be measured directly. The coil carries a constant current of 1.50 A. A magnetic field sensor measures the field strength B at distances x from the centre of the coil along its axis.

The field on the axis is given by B = μ₀NIR²/[2(R² + x²)^3/2], and at the centre by B₀ = μ₀NI/(2R). These can be combined to give (B₀/B)^2/3 = 1 + x²/R². The graph shows six of the seven points.

A small insulated sphere carrying a fixed charge q hangs between two large, vertical, parallel metal plates. The potential difference between the plates is kept at 4.00 kV. A force sensor measures the horizontal electric force F on the sphere. The plates are moved apart in steps, keeping the sphere midway between them, and for each position the student records the reading d_s of a scale fixed to one of the plates.

The student expects that F = qV/d, where d is the separation of the plates, and plots a graph of 1/F against d_s. The graph shows five of the six points.

Two identical small conducting spheres, each of radius 0.50 cm, are given equal charges. One sphere is fixed on an insulating stand; the other is mounted on an insulating rod attached to a digital force sensor, which measures the electrostatic force F between the spheres. The separation r of their centres is set with a scale, and its uncertainty is negligible. Each value of F has an absolute uncertainty of ±0.10 mN.

To test whether F ∝ rⁿ with n = −2, the student plots lg(F / mN) against lg(r / cm). The graph shows five of the six points with their error bars.

An electrostatic precipitator removes soot from the flue gas of a power station. Two vertical parallel metal plates are 0.25 m apart and have a potential difference of 40 kV between them. A soot particle carries a charge of −3.2 × 10⁻¹⁷ C.

An electrostatic precipitator removes soot from the flue gas of a power station. Gas flows between two large parallel metal plates 0.15 m apart. The potential difference between the plates is 30 kV.

A child slides down a plastic playground slide. Afterwards the child carries a charge of −45 nC. Treat the child and the slide as an isolated system that was initially uncharged. The elementary charge is e = 1.60×10⁻¹⁹ C.

Two oil droplets in a food-spraying nozzle carry charges of +2.4 nC and +6.0 nC. Use k = 8.99 × 10⁹ N m² C⁻².

A high-voltage cable crossing a mountain pass carries a steady current of 270 A, flowing from west to east. A walker holds a compass 6.0 m directly below the cable. Use μ₀ = 4π × 10⁻⁷ N A⁻². Treat the cable as a long straight wire.

During a thunderstorm the base of a cloud carries a large negative charge, and the ground below it becomes positively charged. The base of the cloud is 1.2 km above the ground, and the region between them may be modelled as the uniform field between two parallel plates. The electric field strength in this region is 5.0 × 10⁴ V m⁻¹.

Water droplets are spheres of density 1000 kg m⁻³. Viscosity of air = 1.8 × 10⁻⁵ Pa s; density of air = 1.2 kg m⁻³.

Two point charges, A of +4.00 μC and B of −2.00 μC, are fixed 0.300 m apart. Point P is 0.250 m from each charge, and M is the midpoint of AB, as shown.

In electrostatic powder coating, paint powder is blown through a spray gun lined with PTFE (a plastic). The particles are charged by friction as they rub against the lining and leave the gun positively charged. They then travel between a flat metal grid at a potential of +30 kV and a flat earthed metal panel that is to be coated. The grid and the panel are parallel and 0.25 m apart.

A typical powder particle has a mass of 2.0 × 10⁻¹⁰ kg and a charge of +2.4 × 10⁻¹³ C.

A long, straight, horizontal overhead cable of a direct-current power line runs north–south, 10.0 m above level ground. It carries a steady current of 600 A towards the north.

The magnetic field strength at a distance r from a long straight wire carrying current I is B = μ₀I/(2πr), with μ₀ = 4π × 10⁻⁷ T m A⁻¹. The horizontal component of the Earth's magnetic field at the site is 1.8 × 10⁻⁵ T, directed towards geographic north.

In a simple model of the hydrogen atom, the electron (charge −e) moves in a circular orbit of radius 5.29 × 10⁻¹¹ m around a stationary proton (charge +e). Only the electric force between them is considered.