Why Current Seems to Flow Across the Empty Gap of a Charging Capacitor

Your phone, a hospital X-ray machine and the Sun all send out waves made of electric and magnetic fields that sustain each other while crossing empty space. The key that unlocks them is a small correction to Ampere's law.

This lesson covers displacement current, the properties of electromagnetic waves, and the regions of the electromagnetic spectrum.

**Displacement current is the current $I_d={\epsilon }_0\frac{d{\Phi }_E}{dt}$ associated with a changing electric flux, and adding it to conduction current makes Ampere's law hold even where no charge flows.

The problem. While a capacitor charges, a loop around the wire encloses the current I, but a surface through the gap between the plates encloses no conduction current. Ampere's law would then give two different fields for the same loop.

The fix — the Ampere-Maxwell law:**

$$\oint \vec{B}\cdot d\vec{l}={\mu }_0\left(I_c+{\epsilon }_0\frac{d{\Phi }_E}{dt}\right)$$

Inside the gap, the growing electric field supplies exactly the missing current.

Worked example. A capacitor with plates of area 0.020 m${}^2$ is charged by a steady 0.15 A. The field between the plates is $E=Q/{\epsilon }_{0}A$, so

$$\frac{dE}{dt}=\frac{I}{{\epsilon }_{0}A}=\frac{0.15}{8.85\times 10^{-12}\times 0.020}=8.5\times 10^{11}{\text{ V m}}^{-1}{\text{ s}}^{-1}$$

$$I_d={\epsilon }_{0}A\frac{dE}{dt}=8.85\times 10^{-12}\times 0.020\times 8.5\times 10^{11}=0.15\text{ A}$$

The displacement current in the gap equals the conduction current in the wire.

An everyday example. A mobile phone receives signals with no wire at all, because a changing electric field creates a magnetic field, which in turn creates an electric field, carrying the wave through the air.

The substance. Displacement current involves no moving charge — it is a changing electric field that produces a magnetic field just as a real current does.

**Electromagnetic waves are oscillating electric and magnetic fields, perpendicular to each other and to the direction of travel, which move through a vacuum at $c=\frac{1}{\sqrt{{\mu }_0{\epsilon }_0}}$ with $\frac{E_0}{B_0}=c$.

Key characteristics:

- Transverse — E, B and the direction of travel are mutually perpendicular
- No medium needed — they cross the vacuum of space
- In phase — E and B reach their peaks together
- Energy is shared equally between the electric and magnetic fields
- They carry momentum, so they exert radiation pressure on surfaces
- Produced by accelerating charges, such as electrons oscillating in an antenna

Worked example 1 — the speed of light.**

$$c=\frac{1}{\sqrt{4\pi \times 10^{-7}\times 8.85\times 10^{-12}}}=3.0\times 10^8{\text{ m s}}^{-1}$$

Worked example 2. A wave has a magnetic field amplitude of $2.0\times 10^{-8}$ T:

$$E_0=c{B}_0=3.0\times 10^8\times 2.0\times 10^{-8}=6.0{\text{ V m}}^{-1}$$

An everyday example. **Sunlight crosses about $1.5\times 10^{11}$ m of empty space** to reach Earth, taking $\frac{1.5\times 10^{11}}{3.0\times 10^8}=500$ s, a little over eight minutes.

The substance. **A small $B_0$ does not mean a weak magnetic part** — the energy stored in B equals the energy stored in E, even though $B_0$ is c times smaller in SI units.

In order of increasing frequency and decreasing wavelength, the spectrum runs radio waves, microwaves, infrared, visible light, ultraviolet, X-rays and gamma rays, all travelling at c in a vacuum.

Region by region — produced by, detected by, used for:

- Radio waves — oscillating currents in antennas; detected by receiving antennas; radio, television and mobile communication
- Microwaves — special vacuum tubes such as klystrons and magnetrons; point-contact diodes; radar and microwave ovens
- Infrared — hot bodies and molecules; thermopiles and bolometers; remote controls, night vision and physiotherapy heating
- Visible light — electrons changing energy levels in atoms; the eye and photocells; seeing and photography
- Ultraviolet — very hot bodies and special lamps; photocells and photographic film; sterilising water and detecting forged notes
- X-rays — fast electrons striking a metal target; photographic film and Geiger tubes; medical imaging
- Gamma rays — radioactive nuclei; Geiger tubes and ionisation chambers; cancer treatment and sterilising equipment

Worked example. Using $\lambda =c/f$:

$${\lambda }_{\text{FM}}=\frac{3.0\times 10^8}{100\times 10^6}=3.0\text{ m}{\lambda }_{\text{oven}}=\frac{3.0\times 10^8}{2.45\times 10^9}=0.122\text{ m}$$

Green light of wavelength 500 nm has $f=\frac{3.0\times 10^8}{5.0\times 10^{-7}}=6.0\times 10^{14}$ Hz.

An everyday example. At a railway station, FM announcements, the phone in your hand, the sunlight on the platform and the baggage X-ray scanner all use different regions of one spectrum.

The substance. The regions differ only in frequency and wavelength — their boundaries overlap, so a given wavelength can belong to two neighbouring regions depending on how it is produced.