Electromagnetism

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Table of Contents

Scientific Theories

A scientific theory is an axiom system
- designed to explain certain kinds of phenomena
- defined by its postulates
- supported or disproved by its predictions

Theory of Electromagnetism

Electromagnetism is the theory of the electromagnetic force

Big Picture

Fields

Electromagnetism introduced the fundamental idea of a field into physics.

In Classical Mechanics forces act directly and instantaneously, e.g. the gravitational forces exerted on each other by the Sun and Earth.
In Electromagnetism phenomena exert forces on objects indirectly, by way of fields propagating at the speed of light, e.g. the electrical force between positive and negative charges.

In the first half of the 19th century Michael Faraday developed the idea of a field but, with only a rudimentary education, was unable to articulate it mathematically. He could, however, draw his new idea, using lines of force (Figures 1 and 2).
James Clerk Maxwell realized the importance of Faraday’s ideas and took on the project of putting them into mathematical form.

A Field is a physical quantity that has a numeric value at every point in space
The values change over time in a smooth, continuous way.
The values can be
- Scalars (single numbers)
  - E.g. temperature
- Vectors (abstract entities having magnitude and direction)
  - E.g. electric, magnetic, and gravitational fields
- Tensors (abstract entities whose components transform across coordinate systems)
  - E.g. the metric tensor of General Relativity

Electromagnetism postulates two (vector) fields:
- the Electric Field and
- the Magnetic Field.
The fields are represented by either
- Vector Arrows or
- Lines of Force (Field Lines)
The length of a Vector Arrow and the proximity of Lines of Force indicate the strength of a field.
- The longer the arrow, the stronger the field
- The more dense lines of force, the stronger the field.

**Lines of Force for an Electric Field**

The Electric Field exerts a force:
- on a positive charge (e.g. a positive ion) in the direction of the lines of force (or vector arrows)
- on a negative charge (e.g. an electron) opposite to the lines of force

The Magnetic Field is more complicated, making moving charged particles go sideways.

**Youtube Video of a charged particle moving through a magnetic field**

Postulates

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Equation of Motion

Rough Statement
- Forces make particles move
Formula
- F = MA
- that is, A = F/M
English Translation of Formula
- The acceleration A of a particle equals the net force F on it divided by its mass M
Since acceleration is the rate of change of velocity

Since velocity is the rate of change of location.

Using the magic of differential equations, the future location, velocity, and acceleration of a particle can be calculated from its:
- initial (measured) location x
- initial (measured) velocity v
- initial (calculated) acceleration A

Lorentz Force Law

Rough Statement
- Magnetic and electric fields exert forces on charged particles
Formula
- F = qE + qv × B
  - F is force
  - E is the electric field
  - B is the magnetic field
  - q is the electric charge of a particle
  - v is is the velocity of a particle

English Translation of F = qE

English Translation of F = qv × B

Coulomb’s Law

Rough Statement
- An electric charge produces an electric field
Formula
- div E = ρ/ε₀, for any point in space
  - div is the degree of divergence from a point
  - E is the electric field
  - ρ is charge density
  - ε₀ is a physical constant
English Translation of Formula
- An electric field emanates and diverges from any point in space where there is a positive electric charge.
- An electric field converges and disappears into a point in space where there is a negative electric charge.

View Interactive Electric Field for Three Charges

Faraday’s Law of Induction

Rough Statement
- A changing magnetic field produces an electric field
Formula
- curl E = -∂B/∂t, for any point in space
  - curl is the degree of rotation around a point
  - E is the electric field
  - B is the magnetic field
English Translation of Formula
- For a given point, a changing magnetic field at the point produces an electric field rotating around it.
Example
- A bar magnet moving back and forth inside a coil of wire produces a changing magnetic field. By Faraday’s Law this produces an electric field extending through the coil, creating an electric current in the wire.

Ampere-Maxwell Law

Rough Statement
- An electric current or a changing electric field produces a magnetic field. That is:
  - Ampere’s Law
    - An electric current in a wire produces a circular magnetic field around the wire
  - Maxwell’s Law
    - A changing electric field produces a magnetic field.
Formula
- curl B = μ₀J + μ₀ε₀∂E/∂t, for any point in space
  - curl is the degree of rotation around a point
  - B is the magnetic field
  - μ₀ Is a physical constant
  - J is current density
  - ε₀ is a physical constant
  - E is the electric field
English Translation of curl B = μ₀J
- For a given point, an electric current at the point produces a magnetic field rotating around it.

Translation of curl B = μ₀ε₀∂E/∂t
- For a given point, a changing electric field at the point produces a magnetic field rotating around it.

No Magnetic Charge Law

Rough Statement
- There are no magnetic charges.
Formula
- div B = 0, for any point in space
  - div is the degree of divergence from a point
  - B is the magnetic field
English Translation of Formula
- No magnetic field emanates from and disappears into any point in space.
- That is, all magnetic field lines loop.

Compare the diagrams:
- Left: The electric lines of force produced by positive and negative charges
- Right: The magnetic lines of force produced by a bar magnet.

It’s natural to think that the magnetic lines of force emanating from the magnet diverge from a north magnetic charge and converge toward a south magnetic charge, like electric charges. But there are no magnetic charges.
Magnetic lines of force emanate from north and south poles because a bar magnet is an electric current loop, made up of billions of tiny atomic current loops.
A current flows counterclockwise in the loop in the diagram.
Circular magnetic lines of force surround the wire, by Ampere’s Law, resulting in lines entering the loop from below and exiting the loop from above.

Postulates Diagrammed

Predictions

Prediction of Electromagnetic Waves

Maxwell realized that three of his equations, taken together, logically implied a phenomenon completely unknown at the time.
Suppose electrons travel back and forth from one end of a straight wire to the other, speeding up, slowing down, and momentarily stopping.
Ampere’s Law implies that a continuously changing magnetic field surrounds the wire.
By Faraday’s Law of Induction this changing magnetic field produces a changing electric field.
By Maxwell’s Law this changing electric field produces a changing magnetic field.
By Faraday’s Law of Induction …
By Maxwell’s Law …
And so on.

Historical Note: Faraday speculated in 1845 that oscillations in electric and magnetic lines of force would propagate as waves.

The Result is an Electromagnetic Wave

Propagation of a Plane Electromagnetic Wave

The electric wave is bluish. The magnetic wave reddish.

View Animation

Maxwell’s Inference that Light is an Electromagnetic Wave

From his equations Maxwell calculated that the speed of an electromagnetic wave was
- v = √(1/(μ₀ε₀))
- where
  - ε₀ is a physical constant, called the electric permittivity of a vacuum, whose value is determined by Coulomb’s Law (div E = ρ/ε₀)
    - ε₀ = 4π10^-7
  - μ₀ is a physical constant, called the magnetic permeability of free space, whose value is determined by Ampere’s Law (curl B = μ₀J)
    - μ₀ = 8.854 x 10^-12
Doing the math
- v = √(1/(μ₀ε₀)) = 299,800,000 meters per second = the speed of light
Maxwell concluded “The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the field according to electromagnetic laws.”
- A Dynamical Theory of the Electromagnetic Field, 1865

Confirmation of Maxwell’s Prediction

In 1886 Heinrich Hertz confirmed Maxwell’s prediction of electromagnetic waves, by generating and detecting radio waves.

Apparatus:
- Transmitter: a straight electrical wire about a yard long with a gap in the middle for a spark to cross.
  - Attached to the the wire are a capacitor, inductor, and battery making what’s called an LC oscillator.
- Receiver: a curved electrical wire about a yard long with a gap in the middle for a spark to cross.
What Happens
- Electrons moving back and forth in the transmitter wire generate radio waves (and make sparks across its gap).
- The radio waves make electrons move back and forth in the receiving wire, making sparks across its gap.

Brief History

Before 1820 electricity and magnetism were believed to be unrelated phenomena
1820 Hans Christian Ørsted
- Discovered that a magnetic needle aligns itself perpendicularly to a current-carrying wire

1820-1855 Michael Faraday
- From working class family with only a rudimentary education
- At age 14 was apprenticed to a bookbinder
- Discovered Faraday’s Law of Induction
- Made the first electric motor and first electric generator.
- Developed the basic ideas of electromagnetism (without math): lines of force and electric and magnetic fields propagating at a rapid but finite speed.
- Though famous for his ingenious experiments, his contemporaries dismissed his theoretical ideas as obscure, mathless musings.
- Britannica:
  - He provided the experimental, and a good deal of the theoretical, foundation upon which James Clerk Maxwell erected classical electromagnetic field theory.

Faraday’s *Experimental Researches in Electricity*

1841 William Thomson (Lord Kelvin)
- At age 17 proved that Faraday’s lines of force can be represented mathematically by the same Fourier equations that govern the flow of heat in a metal bar

1855-1875 James Clerk Maxwell
- At age 14 published his first paper on mathematics, went to Cambridge.
- Maxwell read Thompson’s paper on Faraday’s lines of force and Faraday’s Experimental Researches in Electricity.
- Made it his mission to convert Faraday’s physical ideas into mathematical form. Developed the mathematical theory of electromagnetism
- Derived from postulates of electromagnetism one of the great predictions of science: electromagnetic waves
- Died at 48 in 1879 before his prediction was tested.
- His contemporaries didn’t understand his theory
  - The concept of a field was too revolutionary
  - The mathematics was new and complex
- Britannica:
  - He is ranked with Sir Isaac Newton and Albert Einstein for the fundamental nature of his contributions.

1880-1890 Maxwellians (Oliver Lodge, GF FitzGerald, Oliver Heaviside, Heinrich Hertz)
- 1885 Oliver Heaviside, having developed vector calculus, reduced Maxwell’s 20 equations to the four known as Maxwell’s Equations
- 1886 Heinrich Hertz tested Maxwell’s great prediction

1905 Einstein
- Published On The Electrodynamics Of Moving Bodies
- When asked if he stood on the shoulders of Newton, Einstein replied “I stood on Maxwell’s shoulders.”

Applications

Electric Generator

Michael Faraday, based on his Law of Induction, made the first electric generator.
You put a coil of wire in a magnetic field and rotate the coil. By Faraday’s Law of Induction, a current is generated in the wire.
The coil can be rotated by hand, by water pressure from a dam, by air pressure from a wind turbine, or by steam pressure from a coal plant or nuclear reactor

Image Credit courses.lumenlearning.com/boundless-physics/chapter/magnetic-flux-induction-and-faradays-law/

Electric Motor

Michael Faraday also made the first electric motor, which is simply an electric generator in reverse.
You put a coil of wire in a magnetic field and and run a current through it. That makes the coil rotate, per the Lorentz Force Law.
That is, electrons moving through a magnetic field experience a sideways force, shown by F in the diagram. The force on upper wire of the coil is straight up and the force on the lower wire is straight down. Combined, the forces turn the coil

Image Credit hades.mech.northwestern.edu/index.php/Brushed_DC_Motor_Theory

Smoke Detectors

Smoke detectors in homes are Ionization Smoke Detectors. Here’s how they work.
In the Left Circuit, with a battery and a chamber of air molecules, no current flows because the electrons in the wire cannot cross the Air Chamber.
The Air Chamber in the Right Circuit has attached to it a supply of the radioactive element Americium-241, which sends alpha particles into the chamber. The alpha particles combine with air molecules to form ions, positively and negatively charged particles. Being charged, the ions carry the current across the chamber, completing the circuit.
But any smoke particles entering the chamber combine with the ions, weighing them down and breaking the circuit. The system is rigged so when the current stops, the alarm goes off.

Electric Shocks

From Electricity and Magnetism, 3rd edition, 2013, Purcell and Morin, page 218
- The severity of a shock depends on the current, not on the applied voltage. Duration also matters.
- An electric current is dangerous because it can cause burns and ventricular fibrillation.
  - Currents as small 50 mA can cause fibrillation.
  - A current of 10 mA running through your hand is the “can’t let go” threshold. Sweating makes things worse, reducing the resistivity of the skin.
- Electrical workers sometimes keep a hand in a pocket, reducing the chance of touching a grounded object and forming a conductive path, perhaps through the heart.
- If you shuffle your feet on a carpet on a dry day, you can charge yourself up to 50,000 V. But the shock you get isn’t lethal because the amount of charge on you is tiny.. But a 120 V wall socket can kill you, since the amount of charge from the power company is essentially infinite.

Ground Fault Circuit Interrupter

Earth is like bar magnet

If the Earth is like a bar magnet with the magnet’s north pole at the North Pole and its south pole at the South Pole, the magnetic lines of force at the Earth’s surface would run from the North Pole to the South Pole, making a magnet indicate that North is to the south.

Except the north end of the bar is (currently) at the South Pole

The geomagnetic south pole is at the geographic North Pole and the geomagnetic north pole is at the geographic South Pole
The poles reverse on average every 300,000 to a million years. Reversals are relatively fast, geologically speaking, about 5,000 years.

Pickups

Electric pianos, violins, violas, cellos, basses, guitars, banjos, and mandolins generate sound using a system of pickups, amplifiers, and speakers.

By Ampere’s Law, a current through the pickup’s coil produces a magnetic field.
By Faraday’s Law of Induction, a steel string vibrating in a magnetic field produces a current in the string.

Image Credit seymourduncan.com/blog/latest-updates/the-anatomy-of-single-coil-pickups

Image Credit: premierguitar.com/maxwells-silver-hammer-of-the-gods