Poynting vector, LC circuits, and standing waves
Electromagnetic fields carry energy. Follow where it flows, watch it slosh back and forth in a circuit, and see it pile up in a standing wave.
Switch between Radiating Source and Current in Wire. For the antenna, energy flows outward; for the wire, it flows in from the surrounding fields.
The Poynting vector S shows the direction of electromagnetic energy flow. For a radiating source, energy flows outward. For a resistive wire, energy flows inward from the fields.
Electromagnetic fields carry energy. The Poynting vector S = (1/μ₀)(E × B) describes the direction and rate of energy flow. LC circuits oscillate energy between electric and magnetic forms. Standing waves create fixed patterns used in lasers and microwave ovens.
The Poynting vector shows how energy flows through electromagnetic fields. For a radiating source, energy flows outward. For a current-carrying wire, energy flows inward from the fields into the wire.
Key insight: Energy does not flow through wires — it flows through the electromagnetic fields surrounding them. The wire guides the fields; the fields carry the energy.
Adjust the inductance L and capacitance C. Watch energy swap between the capacitor's electric field and the inductor's magnetic field, and how the frequency changes.
Energy oscillates between the capacitor (electric field) and inductor (magnetic field). Analogous to a mass-spring system: Q ↔ position, I ↔ velocity, L ↔ mass, 1/C ↔ spring constant.
An LC circuit oscillates energy between the electric field of the capacitor and the magnetic field of the inductor at frequency f = 1/(2π√LC). It is the electromagnetic analogue of a mass on a spring.
Key insight: LC circuits are the basis of radio tuning. By adjusting L or C, you change the resonant frequency, selecting which radio station you receive.
Slide the wavelength and watch two opposing waves add up to a standing wave. The nodes stay put, half a wavelength apart.
When two identical waves travel in opposite directions, they create a standing wave with fixed nodes (zero displacement) and antinodes (maximum displacement) spaced λ/2 apart.
Two counter-propagating waves create a standing wave with fixed nodes and antinodes. Nodes are spaced λ/2 apart. Standing waves are essential in laser cavities, microwave ovens, and musical instruments.
Key insight: In a microwave oven, standing waves create hot spots (antinodes) and cold spots (nodes). The rotating turntable moves food through both to heat it evenly.
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