doPhysics · Electromagnetism 2
Transformers, LC Oscillations and Electromagnetic Waves
Three uses of changing fields. Two coils on one core step voltages up and down and carry power across the country. A capacitor and a coil swing energy to and fro like a block on a spring. Maxwell's displacement current completes the laws of electromagnetism, and they predict waves of electric and magnetic field that travel at the speed of light, from radio waves to gamma rays.
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Why this chapter
Why this chapter exists
The last chapters found that a changing magnetic field makes an electric field, and built coils, generators and alternating currents on that idea. This chapter takes it three steps further. Two coils on one iron core make a transformer, which carries the power of a dam to the box on your lane. A capacitor and a coil make an oscillator, which lets a radio pick out one station. Finally, Maxwell added one term to Ampere's law, and the laws of electricity and magnetism predicted waves. Their speed came out as the speed of light, and light turned out to be one of them.
You will learn
By the end of this chapter you will
- Explain how a transformer works, and use the turns ratio, the current ratio and the efficiency.
- Explain why power is sent at high voltage, and find the loss in a line.
- Describe the LC oscillation, write \(q(t)\) and \(i(t)\), and find \(\omega = 1/\sqrt{LC}\).
- Derive the LC equation and match it term by term with the block on a spring, including the energies.
- Explain the gap in Ampere's law, define the displacement current, and name Maxwell's four equations.
- Describe a plane electromagnetic wave, with \(E\) and \(B\) in phase and the travel along \(E \times B\). Use \(E_0 = cB_0\) and \(c = 1/\sqrt{\mu_0\varepsilon_0}\).
- Find the energy density, intensity and radiation pressure of a wave, and place any wave in the electromagnetic spectrum.
Before you begin
Before you begin
From the chapter Eddy Currents, Generators and Inductance you should know Faraday's law, self-inductance and mutual inductance, and the energy \(\tfrac12 Li^2\) of a coil. From the chapter Capacitance and Dielectrics you should know \(C\), the field between parallel plates and the energy \(q^2/2C\). From the chapter Sources of the Magnetic Field you should know Ampere's circuital law. You should know the block on a spring and its energy from the chapter Dynamics of SHM. You should know a travelling wave \(y = A\sin(kx - \omega t)\) from the chapter Introduction to Waves.
Lessons
1
The Transformer
Two coils on one iron core share one changing flux, so the voltages are in the ratio of the turns and the currents in the inverse ratio. A real transformer loses a little energy in four ways, and power travels at high voltage because the line loss grows as the square of the current.
Mutual induction makes a transformerCurrents in an ideal transformerEfficiency and lossesWhy power travels at high voltage
2
LC Oscillations
A charged capacitor joined across an inductor makes charge and current swing to and fro, a quarter period apart, at \(\omega = 1/\sqrt{LC}\). The equation is the equation of a block on a spring, and the energy passes between the electric field of the capacitor and the magnetic field of the coil while its total stays fixed.
Charge and current trade placesThe lc equation and the shm analogyEnergy in the two fieldsWhy real oscillations die away
3
Maxwell's Displacement Current
Ampere's law gives two answers for one loop round the lead of a charging capacitor. Maxwell's displacement current, \(\varepsilon_0\) times the rate of change of electric flux, removes the contradiction, makes a magnetic field between the plates, and completes the four equations of electromagnetism.
The gap in amperes lawDisplacement currentMaxwells four equations
4
Electromagnetic Waves
A changing magnetic field makes an electric field and a changing electric field makes a magnetic field, so an accelerated charge sends the pair outward as a wave. In a plane wave \(E\) and \(B\) are perpendicular, in phase and transverse, the wave runs along \(E \times B\), \(E_0 = cB_0\), and \(c = 1/\sqrt{\mu_0\varepsilon_0}\) is the speed of light.
How a changing field makes a waveThe plane wave e and b in stepThe speed of light from mu0 and epsilon0
5
Energy, Intensity and Pressure of Light
An electromagnetic wave carries energy, shared equally between its electric and magnetic fields, with average density \(\tfrac12\varepsilon_0E_0^2\). Its intensity is that energy times \(c\), it falls as \(1/r^2\) from a point source, and the wave carries momentum \(U/c\), so light presses on whatever absorbs or reflects it.
Energy density of the waveIntensity of a sourceRadiation pressure
6
The Electromagnetic Spectrum
Radio waves, microwaves, infrared, light, ultraviolet, X-rays and gamma rays are one kind of wave at different frequencies, all travelling at c. Each band is made by its own kind of source and has its own uses, and the ionosphere decides which radio waves go round the Earth and which reach space.
The spectrum in orderHow each band is made and usedRadio television and the ground station
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%The Workbook
Questions to solve on paper, marked by you.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- Ampere-Maxwell law
- Ampere's circuital law with the displacement current added: \(\oint\vec B\cdot d\vec l = \mu_0(I + I_d)\).
- antenna
- A conductor in which charges are driven to and fro to send out electromagnetic waves, or in which arriving waves drive currents.
- damping
- The gradual loss of the energy of an oscillation, so that its amplitude falls. In an LC circuit the energy goes to heat and to radiation.
- displacement current
- Maxwell's term \(I_d = \epsilon_0\,d\Phi_E/dt\): a changing electric flux that makes a magnetic field just as a current of the same size does. No charge flows.
- efficiency
- The fraction of the input power that a device gives out as useful output, \(\eta = P_\text{out}/P_\text{in}\), often written as a percentage.
- electric flux
- The product of an electric field and the area it crosses at right angles, \(\Phi_E = EA\) for a uniform field. In general it is \(\int\vec E\cdot d\vec A\).
- electromagnetic spectrum
- The whole range of electromagnetic waves arranged by frequency: radio, microwave, infrared, visible, ultraviolet, X-rays and gamma rays.
- electromagnetic wave
- A travelling pattern of electric and magnetic fields, each changing field making the other. In vacuum it travels at \(c = 1/\sqrt{\mu_0\epsilon_0}\).
- energy density
- Energy per unit volume. For an electromagnetic wave it is \(\tfrac12\epsilon_0E^2 + B^2/2\mu_0\), with the two halves equal.
- hysteresis
- The lag of the magnetisation of iron behind the field that magnetises it. Taking iron round its hysteresis loop once costs energy, which appears as heat.
- intensity
- The power carried by a wave across unit area held at right angles to its travel, in W/m².
- ionosphere
- A layer of the upper atmosphere, from about \(60\,\text{km}\) up, full of charged particles made by sunlight. It turns back radio waves below about \(30\,\text{MHz}\).
- LC circuit
- A circuit of a capacitor and a coil. Charged and left alone, it makes charge and current oscillate at \(\omega = 1/\sqrt{LC}\).
- magnetising current
- The small current a transformer's primary draws when its secondary is open. It keeps the flux alternating and takes almost no power.
- Maxwell's equations
- The four laws of electromagnetism: Gauss's law for \(\vec E\), Gauss's law for \(\vec B\), Faraday's law and the Ampere-Maxwell law.
- mutual induction
- The induction of an emf in one coil by a changing current in another coil nearby, through the flux they share.
- natural frequency
- The frequency at which a system oscillates when it is disturbed and left alone. For an LC circuit it is \(1/2\pi\sqrt{LC}\).
- photon
- A packet of electromagnetic energy \(hf\), studied in the chapter Photons and the Photoelectric Effect. Its energy grows with the frequency.
- plane wave
- A wave whose wavefronts are flat planes, as a small part of a spherical wave is far from its source.
- primary coil
- The coil of a transformer joined to the supply. Its alternating current makes the alternating flux in the core.
- radiation pressure
- The pressure a wave exerts on a surface because it carries momentum: \(I/c\) if the surface absorbs it, \(2I/c\) if it reflects it.
- resonance
- The large response of an oscillating system driven at its own natural frequency.
- secondary coil
- The coil of a transformer joined to the load. The changing flux of the core induces its voltage.
- step-down transformer
- A transformer with fewer turns on the secondary than on the primary. It lowers the voltage and raises the current.
- step-up transformer
- A transformer with more turns on the secondary than on the primary. It raises the voltage and lowers the current.
- transformer
- Two coils wound on one iron core. An alternating voltage on one coil induces an alternating voltage on the other, in the ratio of their numbers of turns.
- transmission line
- The pair of long wires that carries electrical power from a power station to its users. Its resistance turns a part of the power into heat, \(I^2R\).
- transverse wave
- A wave in which the disturbance is at right angles to the direction of travel. In an electromagnetic wave both \(\vec E\) and \(\vec B\) are transverse.
- turns ratio
- The ratio \(N_s/N_p\) of the numbers of turns on the secondary and the primary of a transformer. For an ideal transformer it equals \(V_s/V_p\).