Chapter 9: Fields

The chapter problem: the spacecraft in low Earth orbit

A small spacecraft is in a low orbit, 400 km above the Earth's surface. The mission team checks its orbit, its ion drive and its power supply.

These are the facts to use all the way through the chapter.

• The spacecraft orbits at a height of 400 km above the Earth's surface. The mass of the Earth is $5.97 \times 10^{24}\,\text{kg}$ and its radius is $6.37 \times 10^{6}\,\text{m}$. The gravitational constant is $G = 6.67 \times 10^{-11}\,\text{N}\,\text{m}^{2}\,\text{kg}^{-2}$.
• The ion drive makes xenon ions. Each ion has a charge of $1.60 \times 10^{-19}\,\text{C}$ and a mass of $2.18 \times 10^{-25}\,\text{kg}$. A potential difference of 1.20 kV accelerates the ions between two grids 1.5 mm apart.
• The ion beam carries a current of 0.50 A.
• The spacecraft trails a conducting tether 20 m long. It moves at the orbital speed at right angles to the Earth's magnetic field. The flux density at the orbit is $3.0 \times 10^{-5}\,\text{T}$. The tether drives a current of 0.80 A.
• The Coulomb constant is $k = \dfrac{1}{4\pi\varepsilon_0} = 8.99 \times 10^{9}\,\text{N}\,\text{m}^{2}\,\text{C}^{-2}$.

By the end of the chapter you will find the gravitational field strength and the gravitational potential at the orbit, and the speed and period of the spacecraft. You will find the electric field between the grids and the speed the ions gain. You will find the magnetic force on the ion beam, and the force that the tether current feels in the Earth's field. You will find the emf induced across the tether. At the end you will decide whether the tether can supply useful power, and explain your reasoning.

First contact: no working yet. Predict. The spacecraft moves at about 7.7 km s⁻¹. The Earth's magnetic field is only $3.0 \times 10^{-5}\,\text{T}$. Will the emf across the 20 m tether be a few millivolts, a few volts, or a few kilovolts? Write down your answer and one sentence of reasoning.

By the end of this chapter you will be able to

• describe a force field, and draw and interpret field lines;
• use Newton's law of gravitation $F = \dfrac{GMm}{r^{2}}$;
• define gravitational field strength and use $g = \dfrac{GM}{r^{2}}$;
• define gravitational potential and use $V = -\dfrac{GM}{r}$;
• explain why gravitational potential is negative, and find the energy needed to move a mass between two points;
• relate field strength to the potential gradient;
• find the speed, the period and the height of an orbit, and the escape speed;
• use Coulomb's law $F = \dfrac{kQq}{r^{2}}$;
• define electric field strength and use $E = \dfrac{F}{Q}$, $E = \dfrac{kQ}{r^{2}}$ and $E = \dfrac{V}{d}$;
• define electric potential and use $V = \dfrac{kQ}{r}$, and find the work done moving a charge;
• compare gravitational fields with electric fields;
• describe the motion of a charged particle in a uniform electric field;
• define magnetic flux density and the tesla;
• use $F = BIL$ and $F = BQv$, and apply Fleming's left-hand rule;
• explain the circular path of a charged particle in a magnetic field, and use $r = \dfrac{mv}{BQ}$;
• define magnetic flux and flux linkage;
• state Faraday's law and Lenz's law, and use $\varepsilon = -N\dfrac{\Delta\Phi}{\Delta t}$;
• find the emf induced in a conductor that moves through a field, $\varepsilon = BLv$;
• describe an alternating-current generator, and use $\varepsilon_0 = NAB\omega$;
• explain how a search coil and a transformer use electromagnetic induction.

Where it is used

A mission planner works out the speed and the period of a satellite orbit. A power engineer follows the potential gradient around a high-voltage insulator. A hospital turns a beam of protons with a magnetic field to aim it at a tumour. A generator spins a coil in a magnetic field to light a city. A bicycle dynamo turns a wheel into a current. An induction hob heats a pan without touching it. A spacecraft trails a tether to generate power from the Earth's magnetic field. The same ideas explain why an apple falls, why a lightning rod protects a building, and why the Earth's magnetic field bends the solar wind around us.

Your turn to connect it

Think of a time when a magnet moved near a wire, or a wire moved near a magnet. It could be a bicycle dynamo, a wireless phone charger, a security tag at a shop door, a tram, or a wind turbine. Write down two examples. For one of them, say what you think the movement does to the charges in the wire.

Roadmap

1. 9.1 Gravitational fields
2. 9.2 Electric fields
3. 9.3 Magnetic fields
4. 9.4 Electromagnetic induction
5. Review