Electromagnetics

Differential volume and area elements in Cartesian, cylindrical and spherical coordinates. Match the scale factors (ρ, r, r sinθ) on curved edges yourself.

A field attaches a value to every point of space. Fill bare points with numbers, then arrows, to see for yourself the difference between a scalar field and a vector field.

The electric field is a vector field a charge creates around it. Use the distance slider and sign toggle to see the 1/r² falloff and direction reversal yourself, and read off the force F = qE on a test charge.

The field of a continuous charge is the vector integral of the tiny field dE from each point-charge piece dq. Add more pieces and watch the sum converge to the integral.

Flux is how much field passes through a surface, counting only the perpendicular component (Φ=E·A=EA cosθ). Tilt the surface to see the cosθ dependence and how the net flux through a closed surface leads to divergence.

The net electric flux through a closed surface is proportional only to the charge inside (∮E·dA=Q/ε₀). See that it is invariant as the Gaussian surface grows, zero when the charge is outside, and that symmetry yields E without integrating.

The net flux through a closed surface equals the volume integral of the divergence inside (∮E·dA=∫∇·E dV). Subdivide into cells to see interior faces cancel and only the boundary remain, and equip the tool that extracts the differential form ∇·E=ρ/ε₀.

The potential V is a scalar field, the potential energy per unit charge (V=kQ/r). Pick an equipotential to see V stay constant and the field cross it perpendicularly, from high V to low V (E=-∇V).

The gradient ∇V extracts the steepest-ascent direction and slope from a scalar field as one vector. Rotate a test direction to see the directional derivative ∇V·û peak when aligned, and equip the tool that yields the field E=-∇V.

In a field a dielectric polarizes; surface bound charge makes an opposing field and the net field weakens by ε_r (E=E₀/ε_r). Raise ε_r to see the dipoles split and the field weaken.

Capacitance is the charge two conductors hold per volt (C=Q/V=εA/d). Narrow the plate gap and insert a dielectric to see that capacitance is set by geometry and material, and grows by ε_r with a dielectric.

Electrostatic energy resides in the electric field itself, not the charges. The density u=½εE² scales with the square of the field, concentrating where it is strong, and the total is U=∫u dV=½CV². Strengthen the field to see the E² growth.

A magnetic field is made by current. The Biot–Savart law gives the tiny field dB=(μ₀/4π)I dl×r̂/r² of a current element. Rotate the test point to see dB perpendicular to both current and distance, growing as sinθ, and that B=∫dB.

The circulation ∮B·dl is proportional only to the current threading the loop (∮B·dl=μ₀I_enc). See it stay invariant as the loop grows, zero when the wire is outside, and get B=μ₀I/2πr without integrating when symmetric.

A magnetic field exerts force on a moving charge. F=qv×B is perpendicular to both v and B, of magnitude qvB sinθ. Change the angle to see the sinθ dependence and learn that it does no work, so the charge traces a circle or helix.

The loop circulation ∮F·dl equals the surface integral of the curl (∮F·dl=∫(∇×F)·dA). Subdivide into cells to see interior edges cancel and only the boundary loop remain, and equip the tool that extracts the differential form ∇×B=μ₀J.

Inductance L is a coil's magnetic inertia against change in its own current (L=NΦ/I). Raise the rate of current change to see the back-emf ε=-L dI/dt grow, and learn the stored energy ½LI².

Atomic magnets in a material align (magnetization M) and the net field grows by μ_r (B=μ_r B₀=μH). Raise μ_r to see the moments align and the field amplify, in contrast to how a dielectric weakened the electric field.

Magnetic energy resides in the magnetic field itself, not the current. The density u=B²/2μ scales with the square of the field, concentrating where it is strong, and the total is U=∫u dV=½LI². Strengthen the field to see the B² growth.

A changing magnetic flux makes an emf (ε=-dΦ/dt). Rotate the loop to see the flux Φ=BA cosθ and the emf ε∝sinθ a quarter cycle out of phase, the flux maximal and the emf zero face-on, and learn the differential form ∇×E=-∂B/∂t.

A changing electric field makes a magnetic field (I_d=ε₀ dΦ_E/dt). Charge a capacitor to see why a magnetic field circles even the gap between the plates, where no charge crosses, and learn the corrected Ampère ∮B·dl=μ₀(I_c+I_d) and the differential form ∇×B=μ₀J+μ₀ε₀∂E/∂t.

Gather the four laws you already learned in one place. Step through Gauss, magnetic Gauss, Faraday and Ampère–Maxwell with buttons to see the structure of two divergences (sources) and two curls (the loop binding them through time), and how the closed system yields the electromagnetic wave c=1/√(μ₀ε₀)=light.

Meshing Faraday and Ampère–Maxwell in empty space gives the wave equation ∇²E=μ₀ε₀∂²E/∂t². Watch the plane wave travel to learn that E⊥B⊥direction of travel (transverse) and the two are in phase, and that the speed stays c=1/√(μ₀ε₀)=light for every wavelength.

An electromagnetic wave carries energy in its direction of travel, and that flow is the Poynting vector S=E×B/μ₀. Polarization is the direction E swings. Rotate the polarization and B stays perpendicular to E, so the energy flow S changes neither direction nor size; see the intensity I=½ε₀cE₀².

At the boundary between two media, light splits into reflection (θ_r=θ₁) and refraction (Snell's law n₁ sinθ₁=n₂ sinθ₂), because light slows to v=c/n in a medium. Raise the angle of incidence to see total internal reflection when, going dense to rare, it passes the critical angle (sinθ_c=n₂/n₁).

When a line is as long as the wavelength, voltage and current run as waves, and their ratio is the characteristic impedance Z₀=√(L/C). Adjust the load Z_L to see the reflection coefficient Γ=(Z_L−Z₀)/(Z_L+Z₀) and the standing wave, and that at a match (Z_L=Z₀) Γ=0, the reflection vanishes and all the power is delivered.

A tool that maps complex impedance to a point on the reflection-coefficient disc. Plot the normalized z=r+jx by Γ=(z−1)/(z+1) and tune the resistance r and reactance x to bring the point to the matched center (z=1, Γ=0). Learn the grid of resistance circles and reactance arcs, and the point circling along the line (one turn per λ/2).

Make a mismatched load look like Z₀ to drive the reflection to zero (Γ=0, SWR=1) and deliver the power in full. Tune the λ/4 transformer impedance Z₁ to find the value Z₁=√(Z₀Z_L) that makes Z_in=Z₁²/Z_L=Z₀, and watch the standing wave on the main line vanish. Complex loads are matched with a stub. The end of the EM track.