Displacement current: the missing current
Ampere's rule says a current makes a magnetic field around it. Now think of a capacitor being charged. Current flows in the wires. But between the plates there is only air or empty space. No charge crosses it.
Yet a magnetic field is found around the gap too. So something in the gap must act like a current. Maxwell found it: the changing electric field between the plates.
He named it the displacement current, Id:
Id = ε₀ × (dΦE/dt)
Here ΦE is the electric flux (field × area) and ε₀ = 8.85 × 10⁻¹² C²/N m². While charging, Id in the gap equals the current I in the wire. So current is "continuous" all the way round.
Ampere–Maxwell law
The total current that makes a magnetic field is I + Id. In words: a magnetic field is made by moving charges and by a changing electric field.
Why it matters
A changing B makes E (Faraday). A changing E makes B (Maxwell). This two-way link lets the fields keep each other going. That is how light and radio waves exist.
How an EM wave is made
A charge at rest makes only an E field. A charge moving at steady speed also makes a B field, but no wave. A charge that accelerates (speeds up, slows down or wiggles) sends out EM waves.
Example: in a radio antenna, electrons move up and down millions of times a second. They send out radio waves of the same frequency.
Features of electromagnetic waves
- They need no medium. They cross empty space (sunlight reaches us through space).
- In vacuum all of them move at the same speed c = 1/√(μ₀ε₀) = 3 × 10⁸ m/s.
- Speed, frequency and wavelength are linked: c = fλ.
- E and B are in phase: they reach their top value at the same place and time.
- The sizes are linked: E₀ = c B₀.
- In a material (like glass) they go slower: v = c/n, where n is the refractive index.
- They carry energy and momentum. So light can push a little on a surface (radiation pressure).
- They are not bent by electric or magnetic fields, because they have no charge.
Transverse nature of EM waves
In a transverse wave, the wiggle is at right angles to the direction the wave moves. For an EM wave moving along x:
- E wiggles along y (up–down).
- B wiggles along z (side to side).
- The wave moves along x.
E, B and the direction of travel are all at 90° to each other. The direction of travel is the direction of E × B.
Everyday proof
Polaroid sunglasses cut glare. They only let through light whose E field wiggles in one direction. Only a transverse wave can be "filtered" like this. Sound (a longitudinal wave) cannot.
The electromagnetic spectrum: radio to gamma
The spectrum is the full family of EM waves, arranged by wavelength. From longest wavelength (lowest frequency, lowest energy) to shortest:
| Band | Wavelength (about) | How it is made | Uses |
|---|---|---|---|
| Radio | > 0.1 m | Electrons moving in antennas | Radio, TV, mobile phones |
| Microwave | 0.1 m to 1 mm | Magnetron / klystron valves | Radar, microwave oven, Wi-Fi, satellite links |
| Infrared | 1 mm to 700 nm | Hot bodies, vibrating molecules | TV remote, night vision, physiotherapy heat, greenhouse effect |
| Visible | 700 nm to 400 nm | Electrons jumping in atoms | Seeing, photography, fibre optics |
| Ultraviolet | 400 nm to 1 nm | Sun, very hot bodies, arc lamps | Killing germs in water, checking fake notes, LASIK eye surgery |
| X-rays | 1 nm to 10⁻³ nm | Fast electrons hitting a metal target | Bone and tooth scans, airport luggage checks |
| Gamma rays | < 10⁻³ nm | Radioactive nuclei, nuclear reactions | Killing cancer cells, sterilising medical tools |
Safety in simple words
Shorter wavelength means more energy per packet. UV can cause sunburn; the ozone layer blocks most of it. X-rays and gamma rays can harm cells, so their dose is kept small.
Try it at home
Point a TV remote at your phone's front camera and press a button. The camera shows a small light flashing. Your eyes cannot see this infrared light, but the camera can. Now put a glass of water in front: it still works. Put your hand in front: it stops. Then use the λ slider in the 3D to find which band the remote uses.
Key formulas and definitions
- I_d = ε₀ dΦ_E/dt
- c = 1/√(μ₀ε₀) = 3 × 10⁸ m/s
- c = f λ
- E₀ = c B₀
- v = c/n (in a material)
Worked examples
1. An FM station broadcasts at 100 MHz. Find the wavelength.
Step 1: c = fλ, so λ = c/f. Step 2: λ = (3 × 10⁸) / (100 × 10⁶) = 3 m. Answer: 3 m (radio band).
2. Yellow light has wavelength 600 nm. Find its frequency.
Step 1: f = c/λ. Step 2: λ = 600 × 10⁻⁹ m = 6 × 10⁻⁷ m. Step 3: f = 3 × 10⁸ / 6 × 10⁻⁷ = 5 × 10¹⁴ Hz.
3. In an EM wave, the peak magnetic field is B₀ = 2 × 10⁻⁸ T. Find the peak electric field.
Step 1: E₀ = c B₀. Step 2: E₀ = 3 × 10⁸ × 2 × 10⁻⁸ = 6 V/m.
4. A wave has λ = 1 nm. Name the band and find f.
Step 1: 1 nm sits at the border of UV and X-rays; it is taken as soft X-rays. Step 2: f = 3 × 10⁸ / 10⁻⁹ = 3 × 10¹⁷ Hz.
5. A 2 μF capacitor is charged so that the voltage across it rises at 5 × 10⁴ V/s. Find the displacement current between the plates.
Step 1: Charge Q = CV, so the current into the plate is I = C dV/dt. Step 2: In the gap, I_d equals this current. Step 3: I_d = 2 × 10⁻⁶ × 5 × 10⁴ = 0.1 A.
6. A parallel-plate capacitor has plate area 0.02 m². The electric field between the plates changes at 5 × 10¹² V/(m s). Find I_d.
Step 1: Φ_E = E × A, so dΦ_E/dt = A × dE/dt = 0.02 × 5 × 10¹² = 10¹¹ V m/s. Step 2: I_d = ε₀ dΦ_E/dt = 8.85 × 10⁻¹² × 10¹¹ ≈ 0.885 A.
7. Light of frequency 5 × 10¹⁴ Hz enters glass with n = 1.5. Find its speed and wavelength inside the glass.
Step 1: v = c/n = 3 × 10⁸ / 1.5 = 2 × 10⁸ m/s. Step 2: Frequency does not change in a new medium. Step 3: λ = v/f = 2 × 10⁸ / 5 × 10¹⁴ = 4 × 10⁻⁷ m = 400 nm.
8. Check that 1/√(μ₀ε₀) gives the speed of light. (μ₀ = 4π × 10⁻⁷, ε₀ = 8.85 × 10⁻¹²)
Step 1: μ₀ε₀ = 4π × 10⁻⁷ × 8.85 × 10⁻¹² ≈ 1.112 × 10⁻¹⁷. Step 2: √(1.112 × 10⁻¹⁷) ≈ 3.33 × 10⁻⁹. Step 3: 1 / 3.33 × 10⁻⁹ ≈ 3 × 10⁸ m/s. This matches light, so light is an EM wave.
Common mistakes
- Thinking displacement current is a flow of charges. No charge moves across the gap; it is a changing electric field.
- Saying EM waves need a medium like sound does. They travel through vacuum too.
- Mixing up the order of the spectrum. Wavelength goes down (and frequency goes up) from radio to gamma.
- Thinking frequency changes when light enters glass. Speed and wavelength change; frequency stays the same.