§6 · field equations and field components

Reading a graph

How does a curve show the relationship between two physical quantities?

The horizontal axis (abscissa) typically shows the independent variable, like position x or elapsed time t. The vertical axis (ordinate) shows the quantity that depends on it, like particle count or probability density p(x).

A flat horizontal line means the quantity does not change across the axis. A rising curve has positive slope (increasing rate), while a symmetric bell-shaped curve shows high concentration in the centre with tails tapering toward zero on both sides.

One worked example

1. Plot particle positions −3, −1, +1, +3 along the horizontal axis with a vertical bar for particle count.
2. The graph shows symmetric spread centered at zero.
3. The average position is 0, but the points are visibly separated from the origin.

A stopping point: Check axis labels, units, and whether the curve represents a count, a density, or an accumulated total.

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Fractions and ratios

How does dividing into parts compare two quantities?

A fraction a/b compares quantity a to quantity b, or tells us how many parts of size 1/b make up a. When b is not zero, dividing a by b calculates how many units of b are contained within a.

In physical science, a ratio often carries units: dividing displacement in micrometres by time in seconds produces a rate in micrometres per second. A dimensionless ratio occurs when both quantities share the same units, such as the fraction of an ensemble that moves in a given direction.

One worked example

1. If 1 particle out of 4 moves by 3 units, the proportion is 1/4 = 0.25.
2. If the ensemble doubles to 8 particles and 2 move by 3 units, the proportion is 2/8 = 0.25.
3. Multiplying both the count and the total by 2 preserves the underlying ratio.

A stopping point: A ratio compares relative magnitudes; always state whether the physical units cancel or combine.

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A sign records direction

How can movement add up to zero?

Put the starting point at zero on a ruler. A final mark three units right has displacement +3; one three units left has displacement −3. Adding the signed displacements gives zero. Adding the distances from the start gives six.

Displacement compares the final and initial positions. It is not the length of the path travelled between them.

One worked example

Two walkers both finish one unit from their start, one at −1 and the other at +1. The signed average is zero. The average distance is one.

A stopping point: The sign names a direction relative to a chosen axis; it does not mean a negative distance.

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Squares and square roots

Why does four times a mean square mean twice the RMS?

Both +3 and −3 have square 9. Squaring therefore removes the sign while giving greater weight to a larger magnitude. A square root returns the result to the original kind of unit: the square root of a square micrometre is a micrometre.

One worked example

The mean square of −3, −1, +1, +3 is 5. Doubling each displacement gives −6, −2, +2, +6, whose mean square is 20. The RMS changes from approximately 2.236 to 4.472: twice as large, not four times as large.

A stopping point: Use the nonnegative square root when reporting a magnitude.

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Adding and averaging

What does an average keep, and what does it lose?

Imagine four equally sized containers holding 3, 1, 1 and 3 units. Combining them gives 8 units. Sharing the total equally between the four containers gives 2 in each. Two is the average; the original containers were not all equal.

The number of observations and the sum of their values are different quantities. Doubling the number of repeated observations doubles the sum but leaves the average unchanged.

One worked example

1. Add 3 + 1 + 1 + 3 to obtain 8.
2. Count four values.
3. Divide 8 by 4 to obtain 2.
4. Repeating the list gives 16 divided by 8, still 2.

A stopping point: An average is a total shared equally among the number of observations.

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Rates of change and derivatives

What physical rate does a derivative measure?

When a quantity changes over an interval of time or space, the ratio of the change in output to the change in input gives an average rate. As the measurement interval shrinks toward zero, this average ratio approaches a definite limit: the instantaneous derivative.

Geometrically, the derivative is the slope of the tangent line to the function graph at a single point. Physically, it always carries the units of the output divided by the units of the input. For position over time, the derivative is velocity in metres per second; for concentration over position, it is a spatial gradient in particles per metre to the fourth power.

In 1905, Einstein used derivatives to relate microscopic flux to macroscopic concentration gradients, and to extract temperature and wavelength dependencies from radiation laws without guessing unmeasured intermediates.

One worked example

If position x(t) = c t^2 with c = 3 metres per second squared, the change between t and t + Δt is c(t+Δt)^2 - ct^2 = 2ctΔt + c(Δt)^2. Dividing by Δt gives 2ct + cΔt. In the limit Δt -> 0, the instantaneous velocity is exactly 2ct = 6t metres per second.

A stopping point: A derivative is an instantaneous rate bearing explicit units; it is not a fraction of two separate isolated zeros.

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Density is not probability

What does the height of a probability curve mean?

Divide a ruler into bins. The fraction of observations in a bin estimates its probability. Dividing that fraction by the bin width estimates a density. Changing the ruler from metres to micrometres changes the numerical density but not the probability of the same physical interval.

A continuous distribution gives probability zero to any single exact position. At the starting time the ideal point source is instead an atom of probability one, not a finite curve with infinite height.

One worked example

A uniform density of 0.25 per micrometre on a four-micrometre interval has total probability one. A one-micrometre subinterval has probability 0.25; a two-micrometre subinterval has probability 0.5.

A stopping point: Always distinguish curve height, interval area and an individual observation.

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Forces on charges, currents, and electromagnetic waves

How do charges interact with electric and magnetic fields?

Electric charge is the intrinsic property of matter that produces and responds to electromagnetic fields. A charge q at rest in an electric field E experiences an electrostatic force F = q E. When moving with velocity v through a magnetic field B, the charge experiences an additional perpendicular magnetic force F = q (v x B). Together, these form the Lorentz force.

Moving a charge through an electric potential difference Delta V transfers potential energy Delta W = q Delta V. For a fundamental electron charge e = 1.602e-19 C accelerated across a potential of 1 V, the energy gained is defined as 1 electron-volt (1 eV = 1.602e-19 J).

In microscopic matter, an electron bound to an equilibrium position by a restoring force acts as a harmonic oscillator or resonator. When disturbed by incoming electromagnetic waves, it absorbs and reradiates energy at its natural resonant frequency nu_0. Einstein's 1905 light-quanta paper opened with Planck's model of such resonators in thermal equilibrium with radiant energy.

Maxwell's electrodynamics expresses four physical laws in words: electric charges act as sources of electric flux; magnetic field lines are closed loops with no isolated magnetic charges; a time-varying magnetic field induces a circulating electric field (Faraday induction); and electric currents alongside time-varying electric fields generate circulating magnetic fields (Maxwell-Ampere law). Combined, these equations govern self-propagating electromagnetic waves traveling at speed c in vacuum.

Einstein pointed out in 1905 that classical electrodynamics treated the relative motion of a magnet and a conductor with an artificial asymmetry: moving the magnet created an electric field in space that drove current, whereas moving the conductor created no electric field but rather a magnetic Lorentz force on electrons. Yet the physical current was identical in both descriptions.

One worked example

Accelerating an electron of charge e = 1.602e-19 Coulombs across an electric potential difference of Delta V = 1.0 Volt gives kinetic energy Delta W = (1.602e-19 C)(1.0 V) = 1.602e-19 Joules = 1 eV. An electron bound with effective spring constant k_s and mass m oscillates at natural frequency nu_0 = (1 / 2 pi) sqrt(k_s / m).

A stopping point: Charge is the property that makes electric forces; fields mediate force between separated charges without action-at-a-distance.

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Fields, continuous waves, and harmonic functions

What is a continuous wave field and how is its energy measured?

A field assigns a definite physical quantity to every location in space and time. A scalar field assigns a single number, such as temperature or particle density, while a vector field assigns a magnitude and direction, such as electric field strength E or magnetic field B.

Harmonic waves propagate oscillating field values through space according to the phase factor kx - omega t, where the wavenumber k = 2 pi / lambda relates to wavelength lambda and angular frequency omega = 2 pi nu relates to cyclic frequency nu. The wave crests travel at the phase speed c = lambda nu.

Because optical frequencies oscillate hundreds of trillions of times per second (for example, 600 THz corresponds to green-cyan light with wavelength lambda = c / nu = 499.65 nm), measuring instruments record the time-averaged energy flux rather than the instantaneous field oscillation. Over any complete period, the average of cos^2(omega t) is exactly 0.5.

When an isotropic source emits total power P into three dimensions, the energy spreads evenly over concentric spherical wavefronts of area 4 pi r^2. The intensity at distance r is given by I = P / (4 pi r^2). For a 1 W point source, the intensity is 0.0795775 W/m^2 at distance r = 1 m, and drops to one-quarter (0.0198944 W/m^2) at distance r = 2 m.

The Doppler shift in sound is an illustrative analogy for wave frequency shifts: sound waves travel through a material medium (air), which creates an asymmetry between a moving source and a moving listener. This analogy has a strict physical limit: light requires no material ether medium, and in 1905 Einstein demonstrated that electromagnetic wave transformation depends purely on relative velocity between observers.

One worked example

At frequency nu = 600 THz (6.0e14 Hz) with light speed c = 299792458 m/s, the wavelength is lambda = c / nu = 499.65 nm. A 1 W source produces intensity I = 1 / (4 pi (1)^2) = 0.0795775 W/m^2 at 1 m, and I = 1 / (4 pi (2)^2) = 0.0198944 W/m^2 at 2 m.

A stopping point: A field is a value assigned to every place; observed optical intensity is a time average, not an instantaneous pulse.

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Counting what crosses a boundary

Why does a difference of flows change the density?

Take a short interval. If more tracers enter through its boundaries than leave, its count increases. Flux counts signed crossings per area per time; number density counts particles per volume. They are not the same quantity.

In the one-dimensional description, the same balance applies per unit transverse area. The minus sign says an increasing rightward flux removes more particles at the right boundary than arrive at the left.

One worked example

If seven particles enter and five leave during a time interval, the count rises by two. If both flows are five, the count is unchanged even though particles keep crossing.

A stopping point: Conservation counts net crossings; it does not require each individual particle to stop.

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Events, reference frames, and the coordinate grid

How do observers assign spatial and temporal coordinates to physical events?

An event is a physical occurrence localized at a single location in space and a single instant in time, such as a spark jumping, a particle collision, or the emission of a flash of light.

To measure events quantitatively, observers construct a reference frame: an imaginary rigid spatial lattice equipped with stationary clocks at every grid intersection. An observer does not rely on looking at a distant clock across space (which incurs light-travel delay); instead, the coordinate time of an event is recorded by the local clock located right where the event occurs.

Clock synchronization follows Einstein's operational convention: a light pulse sent from clock A at time t_A reaches clock B, reflects, and returns to A at time t'_A. Clock B is synchronized with clock A if its reading at reflection satisfies the midpoint assignment t_B = (t_A + t'_A) / 2.

One worked example

If a master clock A emits a light signal at t_A = 0 seconds toward stationary clock B, and the reflected signal returns to clock A at t'_A = 10 seconds, the synchronization convention assigns the time t_B = (0 + 10) / 2 = 5 seconds to the reflection event at clock B.

A stopping point: An event is something that happens at one place and one time; coordinate assignment is an operational measurement convention, not an absolute cosmic background.

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Functions and graphs

How does a mathematical function record relationships between physical quantities?

A physical function is an unambiguous rule that assigns to each value of an independent input (such as elapsed time t or coordinate position x) a specific value of a dependent quantity (such as particle displacement, local concentration, or field energy). It describes how one aspect of a physical system responds when another changes.

Plotting a function creates a curve on a coordinate plane. The horizontal axis represents the input, and the vertical axis represents the output. Every point on the curve represents a simultaneously paired measurement. Because both axes correspond to physical measurements, any slope or area derived from the graph carries the combined dimensions of those axes.

Distinguishing functional laws from empirical scatter is essential. A theoretical function predicts the ideal expectation of an ensemble, such as the mean square displacement growing linearly with time. Individual experimental trials fluctuate around this expectation, but the functional form governs the ensemble average.

One worked example

For a diffusion coefficient D = 0.5 square micrometres per second, plotting mean square displacement against time t produces a straight line through the origin with slope 2D = 1.0 square micrometres per second. At t = 1 second the value is 1.0 square micrometre; at t = 4 seconds it is 4.0 square micrometres.

A stopping point: A graph is a record of paired quantities with physical units; a slope without units is not an explanation.

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Adding continuously

What operation does an integral describe here?

Approximate the area under a curve by rectangles. Each contributes height times width. Add all contributions, then refine the widths. When those sums approach a limit, that limit is the integral.

For a probability density, a rectangle has units of inverse length times length, leaving a dimensionless probability. For total probability the whole area must be one.

Integration by parts comes from adding the product rule for differentiation over an interval: the integral of u times the derivative of v equals the endpoint product minus the integral of v times the derivative of u.

One worked example

A constant density of 0.25 per micrometre over 4 micrometres gives 0.25 times 4 = 1, regardless of how many equal rectangles we use.

A stopping point: The width and the endpoint term are part of the calculation, not decorations on the integral sign.

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Momentum, energy flux, and radiation pressure of light

How does a beam of light carry energy and momentum?

In classical electromagnetic theory, light is a continuous wave that transports both energy and linear momentum through space. The Poynting vector (introduced by John Henry Poynting in 1884) describes the directional energy flux density in watts per square metre. Because electromagnetic waves travel at the speed of light c, the momentum p carried by a beam is directly proportional to its total energy E through the relation p = E / c.

When a beam of light shines on a target, the absorption of radiant power P delivers a continuous physical force F = P / c. For 1.0 Watt of absorbed light, the mechanical push is 1 / c = 3.33564e-9 Newtons per Watt. When the target acts as an ideal mirror, the light reverses direction and imparts twice its incident momentum, producing a force of 2P / c = 6.67128e-9 Newtons per Watt.

This radiation pressure was experimentally verified with high precision around the turn of the century by Pyotr Lebedev (1901) and Ernest Fox Nichols and Gordon Ferrie Hull (1901-1903), demonstrating that classical electromagnetic fields exert measurable macroscopic mechanical force.

In Einstein's mass-energy paper of September 1905, radiation momentum plays a crucial role: a body at rest that emits two equal pulses of light of energy L/2 in opposite directions imparts equal and opposite momenta (+L / (2c) and -L / (2c)), leaving the emitting body at rest with zero total transferred momentum.

One worked example

A continuous laser beam absorbing 1.0 Watt of power at normal incidence delivers F = (1.0 W) / (299792458 m/s) = 3.33564e-9 N of force. If the surface reflects the entire 1.0 Watt, the force is 2(3.33564e-9 N) = 6.67128e-9 N.

A stopping point: Light's momentum is the push a beam gives when absorbed or reflected; it follows from Maxwell's electrodynamics without requiring quantum hypotheses.

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Partial derivatives and held-fixed quantities

What does a partial derivative mean when several variables change simultaneously?

Many physical quantities depend on more than one parameter: particle density depends on both position x and time t; gas entropy depends on both volume V and temperature T. When asking how such a quantity changes, one must specify which variable is moving and which variables are being held constant.

The partial derivative ∂f/∂x represents the rate of change of f with respect to x while holding time t strictly fixed. Conversely, ∂f/∂t represents the rate of accumulation at a fixed position x over time. In thermodynamics, holding temperature fixed produces an isothermal derivative, while holding entropy or volume fixed produces an adiabatic or isochoric derivative.

Einstein's derivation of the diffusion equation equates the time rate of accumulation at a fixed location to the divergence of spatial flux: ∂f/∂t = D ∂^2f/∂x^2. Both sides describe rates under different held-fixed constraints.

One worked example

Example 1 (Time fixed when moving through space): For diffusion profile f(x, t) = (4 pi D t)^(-1/2) exp(-x^2 / (4Dt)), the spatial gradient quantifies concentration variation along a channel. Here, elapsed time t is strictly fixed as the held-fixed parameter. A spatial snapshot taken at one fixed instant evaluates to ∂f/∂x = -x/(2Dt) f(x, t).

Example 2 (Position fixed when tracking time): The time accumulation rate records concentration changes at a fixed location. Here, spatial position x is strictly fixed as the held-fixed parameter. A stationary probe at one coordinate records accumulation rate ∂f/∂t = (-1/(2t) + x^2/(4Dt^2)) f(x, t). Equating accumulation to spatial flux divergence yields ∂f/∂t = D ∂^2f/∂x^2.

Example 3 (Thermodynamic derivatives: isothermal versus adiabatic): In gas thermodynamics, the volume response to pressure depends on thermal boundary conditions. The isothermal derivative (∂V/∂p)_T explicitly names temperature T as the held-fixed quantity while heat exchanges freely with a bath. Conversely, the adiabatic derivative (∂V/∂p)_S explicitly names entropy S as the held-fixed quantity under thermal insulation. Because isothermal compression permits heat release, the isothermal compressibility exceeds the adiabatic compressibility. Writing ∂V/∂p without identifying the held-fixed parameter is physically incomplete.

Example 4 (Radiation entropy derivative in Light Quanta §3): For blackbody radiation at frequency nu, the entropy density s_nu depends on both spectral energy density u_nu and enclosure volume V. Einstein's derivation of Wien's displacement law takes the partial derivative (∂s_nu/∂u_nu)_(V, nu) = 1/T. Here both cavity volume V and radiation frequency nu are strictly held fixed as the held-fixed parameters while varying energy density u_nu.

Example 5 (Transformed derivatives in Relativity §6): In coordinate transformations between a resting frame (x, y, z, t) and a moving frame (x', y', z', t'), the partial derivative (∂/∂x)_(y, z, t) explicitly holds resting coordinates y, z, and time t fixed. When expressed in moving coordinates via the relativistic chain rule, it becomes a combination of moving spatial derivative (∂/∂x')_(y', z', t') holding y', z', and t' fixed, and moving time derivative (∂/∂t')_(x', y', z') holding x', y', and z' fixed. Changing coordinate systems changes which physical quantities are held fixed.

A stopping point: A partial derivative is mathematically and physically undefined until every held-fixed parameter is explicitly identified.

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Energy of motion and inertia

What can a change in energy of motion tell us when speed stays the same?

Work transfers energy when a force acts through a displacement. The energy associated with the motion of a body is called kinetic energy. It is not the same as the energy of its internal heating, chemistry, or other stored processes.

For the same body at ordinary slow speeds, doubling speed multiplies its energy of motion by four. At a fixed speed, doubling inertial mass doubles that energy. The word inertia describes resistance to a change of motion, not a measurement of gravitational weight.

The formula is the low-speed rule. It cannot be assumed exact for a traveler moving at a substantial fraction of light speed. To identify inertia in a relativistic comparison, examine the coefficient as speed approaches zero.

One worked example

Compare two bodies at 2 metres per second. In the Newtonian model a body of 3 kilograms has 6 joules of energy of motion; a body of 2 kilograms has 4 joules. The speed is unchanged, but the energy of motion differs by 2 joules.

This example assumes independently specified masses; it is not evidence for mass–energy equivalence. The mass-energy argument instead calculates an energy difference from emitted light and then identifies its low-speed coefficient.

A stopping point: Less energy of motion at the same low speed means a smaller inertial mass within the Newtonian approximation. The comparison alone does not supply an absolute internal energy.

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