Electromagnetism and Engines
Coulomb’s Law
Electric charge is the property of matter responsible for electromagnetic phenomena. Properties:
- Interaction can be positive or negative.
- The total charge of a set of particles is the sum with the sign of their individual charges.
- The total electric charge of an isolated system is conserved.
- Charge is quantized: it comes in discrete amounts that are multiples of an elementary amount.
The unit of charge in the SI is the Coulomb (C). Coulomb’s law describes the interaction between electric charges at rest. It states: The force exerted by a point charge q1 on another point charge q2 is proportional to the product of the charges and inversely proportional to the square of the distance, r, separating them. This force is central, directed along the line connecting the charges. It is repulsive if the charges have the same sign and attractive if they have opposite signs.
Formula: F = K ⋅ q1 ⋅ q2 / r2, where K is Coulomb’s constant = 9×109 N⋅m2/C2. In media other than vacuum, K takes other values. The force exerted by q2 on q1 is equal in magnitude and opposite in direction. Electrostatic forces adhere to the principle of superposition: the net force exerted by a set of charges on another is the vector sum of all forces exerted on it.
Electric Potential Energy and Electric Potential
As the force between two charges is conservative, it has an associated electric potential energy function, Ep. The difference in Ep between two points corresponds to the work done by the electric force between those two points: Ep = K ⋅ q1 ⋅ q2 / r, where the electric potential energy at infinity is taken as zero. Like any energy, it is a scalar quantity whose SI unit is the Joule. Under the sole action of the electric force, charges move to positions corresponding to a minimum electric potential energy configuration. The total potential energy of a set of charges is the sum of the potential energies of all possible pairs of charges.
The electric field is also conservative and has an associated scalar field called electric potential. The electric potential produced by a point charge q is: V = K ⋅ q / r. In the SI, potential is measured in volts (V). The potential difference between two points is also called voltage. The potential due to a set of charges is the scalar sum of the potentials due to each of the charges.
Electric Dipole
A physical system formed by two charges of equal magnitude but opposite sign, ±q, and separated by a fixed distance, d. It is characterized by a vector quantity called the dipole moment, whose magnitude is μ = q⋅d.
Ampere’s Theorem
The line integral of a conservative field along a path depends only on the start and end points (closed path integral = 0). The magnetic field is not a conservative field, and its circulation does depend on the path (closed path integral ≠ 0). Ampere’s theorem allows us to calculate the circulation of B along a closed line: “The circulation of the magnetic field strength vector along a closed line is proportional to the intensity of the current passing through the area enclosed by that line. The constant of proportionality is the magnetic permeability of the medium.”
Ampere (Unit)
Current flowing through two parallel conductors of infinite length, separated by a distance of one meter in a vacuum, produces a force on each conductor of F = 2×10-7 N/m.
Faraday and Henry’s Law of Induction
After conducting numerous experiments with magnets and coils, Faraday and Henry reached the following conclusion: when a magnet and a coil move relative to each other, an electric current is induced in the coil. This is called electromagnetic induction. Induced currents are attributed to changes in magnetic flux through the surface of a circuit. These variations may be due to:
- A change in the magnitude or direction of the magnetic field vector (B)
- A change in the magnitude or direction of the surface vector (S)
- Simultaneous variations of both vector quantities.
Faraday-Henry-Lenz’s Law
Any variation of magnetic flux through a closed circuit produces an induced current therein. The induced current is transient and only lasts for the duration of the flux variation. The electromotive force (emf) induced in a circuit (ε) is equal to the negative rate of change of magnetic flux (Φ) passing through it. The direction of the induced current is such that it opposes the change in flux that produces it. These statements can be written by the Faraday-Lenz equation:
ε = -dΦ/dt
(If the flux is expressed in Webers and time in seconds, the emf is given in volts).
One of the main applications of electromagnetic induction is the industrial production of electrical energy. Electromagnetic induction transforms mechanical energy into electrical energy. Power generators use rotating coils within a magnetic field. As the coils rotate, the flux through them changes, inducing an electric current.
AC Generator
By rotating a coil in a magnetic field, the flux varies with time, producing an induced current. In its simplest form, an AC generator consists of a coil that rotates within a magnetic field. Both the magnetic field and the area of the loop remain constant. As the loop rotates, the direction of the magnetic field relative to the loop changes, and the magnetic flux through it changes with time, inducing an electromotive force. If there is an external circuit, a current will flow. The emf that appears in the loop is a sinusoidal function, alternately changing polarity. The frequency of the electric current supplied by electricity companies is typically 50 Hz. For a generator to work, an external energy source (thermal, hydro, nuclear, etc.) is needed to make the coil rotate at the desired frequency. If the frequency is 50 Hz, the current changes direction one hundred times per second. The change happens so quickly that the intensity of light generated in a light bulb appears constant.
Ampere’s Law (Formal)
The circulation of B along a closed curve is equal to the permeability times the current crossing the curve. The line integral of B along a closed curve is not zero, meaning it is not conservative, in contrast to the gravitational or electric fields.
Faraday-Henry-Lenz’s Law (Formal)
When a time variation in magnetic flux occurs, an emf is induced that opposes the change in flux.
Lenz’s Law
When a change in flux occurs, the system opposes the change.
Maxwell’s Laws
- Gauss’s law for the electric field
- Gauss’s law for the magnetic field
- Ampere’s law for B
- Faraday-Henry-Lenz’s law
Engine
An engine is a machine that converts the chemical energy present in fuel into mechanical energy available at its output shaft. In a block diagram of inputs and outputs, the inputs would be air, fuel, and auxiliary systems (lubrication, cooling, and electricity). Inside the engine, there is a distribution system, a piston-crank mechanism, and the output is mechanical energy. Waste products include combustion gases and heat released to the environment.
Differences Between Electric and Magnetic Fields
The magnetic field is a region of space in which a moving electric charge experiences a force perpendicular to both its velocity and the magnetic field. This force is proportional to the charge, velocity, and a property called magnetic induction. The existence of a magnetic field is evidenced by its ability to orient a magnetometer (a magnetic steel needle that can rotate freely). A compass needle, which reveals the existence of the Earth’s magnetic field, can be considered a magnetometer. Unlike the electric field, there are no magnetic monopoles, only magnetic dipoles. This means that magnetic field lines are closed; the net number of field lines entering a surface equals the number of field lines emerging from the same surface. A clear example of this property is the field lines of a magnet, where the same number of field lines leaving the north pole return through the south pole.