Intuition The one core ideaThe whole topic says: a changing electric field acts like an electric current — it makes a magnetic field even where no charge flows. To understand that sentence, you must first know what a field is, what "flux" measures, what a "current" really is, and how we add up things along loops and across surfaces.
This page assumes nothing . Every symbol used by the parent topic is built here from the ground up. Read it top to bottom — each idea is the brick the next one stands on.
A field is a rule that gives you an arrow (a value + a direction) at every point in space . Think of the wind: at each spot in a room the air pushes in some direction with some strength. A field is the map of all those little arrows.
The electric field E
The little arrow over the letter, E direction , not just a size.
Its size (how long the arrow is) is written E
The parent topic's entire claim is about a changing E . Before we can talk about it changing , we must picture the arrows themselves.
The magnetic field B loops rather than pointing away from a source.
Definition Electric chargeQ Q coulombs (C) . Extra electrons piled up = negative Q Q
I Current is the rate at which charge flows past a point. If d Q d t I = d t d Q amperes (A) , where 1 A = 1 coulomb per second.
Read d t d Q "how fast is Q The letter d a tiny change in it , and one thing over another as d ( that ) d ( this ) rate — how much this changes per unit of that . (This is the derivative ; we use it because "rate of change" is exactly the question we keep asking.)
Worked example Feel the unitsIf 3 coulombs flow past in 2 seconds at a steady rate, I = 2 s 3 C = 1.5
Definition Two kinds of current we will nameBecause this topic compares real flowing charge against a changing field pretending to be current , we give them separate names now so no symbol surprises you later:
Conduction current I c — actual charge flowing through a wire, I c = d t d Q c Displacement current I d — the magnetic effect of a changing electric field , defined in §5 once we have flux. No charge moves; the subscript d
When we later ask "how much current pierces a surface?" the answer may be conduction, displacement, or both. We call whichever amount actually threads a chosen surface the enclosed current , written I e n c e n c I e n c role a current plays, while I c I d what kind it is.
The paradox in the parent note is "current pierces one surface but not another." You cannot judge that without knowing current = flowing charge , and that no charge flows across the vacuum gap of a capacitor — so the enclosed conduction current I e n c = I c
Before "flux" we need two smaller ideas.
A d A area vector is an arrow that (i) is perpendicular to the patch and (ii) has a length equal to the patch's area. A tilted window has an area vector poking straight out of the glass.
E ⋅ A dot product of two arrows measures how much they point the same way , scaled by their lengths:
E ⋅ A = E A cos θ θ cos θ 1 0 − 1
Why the dot product and not plain multiplication? Because we care about the part of E through the surface. If E along the surface (angle 9 0 ∘ cos 9 0 ∘ = 0
Flux counts how many field arrows pierce a surface. For a uniform field E Φ E = E ⋅ A = E A cos θ θ = 0 Φ E = E A Φ E of the electric field .
For a bumpy surface or non-uniform field, we chop the surface into tiny patches d A E ⋅ d A add them all up . That "add up infinitely many tiny pieces over a surface" is written with an integral sign:
Φ E = ∫ E ⋅ d A
Intuition Read the integral out loud∫ E ⋅ d A "sweep across the whole surface, and at each tiny patch add the amount of E The ∫
Before we can build the displacement current we need one "exchange-rate" constant.
ε 0 ε 0 ≈ 8.85 × 1 0 − 12 F m − 1 C 2 N − 1 m − 2 how strongly electric fields respond to charge in vacuum: it is the conversion factor between "amount of charge" and "amount of electric field."
Now we can finish the definition promised in §2:
Definition Displacement currentI d I d = ε 0 d t d Φ E "epsilon-nought times how fast the flux is growing." No charge moves — it is a changing electric flux that produces a magnetic field just like a real current would. Flux is the quantity that changes; ε 0
This is the single new object the whole topic is about. Everything else on this page exists so that this one formula reads as plain English.
Definition Closed line integral∮ walk all the way around a closed loop and return to the start . Along the way we chop the loop into tiny steps d l
∮ B ⋅ d l
At each step we take B ⋅ d l how much the magnetic field points along our step — and add them all up around the loop.
Intuition What this number meansIt measures how much B circulates — how strongly it swirls around the loop, like measuring how much a river current pushes you as you paddle once around a circular path. A field that loops around a wire gives a big value; a field that just sits still gives zero.
We also need the second "exchange-rate" constant to connect this circulation to current.
μ 0 μ 0 = 4 π × 1 0 − 7 H m − 1 ≈ 1.2566 × 1 0 − 6 N A − 2 how strongly magnetic fields respond to current : the conversion factor between "amount of current" and "amount of magnetic field."
Ampère's law is exactly the circulation above tied to current by μ 0 ∮ B ⋅ d l = μ 0 I e n c I e n c two different answers for the same loop. You cannot see the paradox without reading this symbol.
There is one hidden agreement we must fix, or the signs go wrong. The loop integral ∮ B ⋅ d l direction of travel around the loop, and the flux Φ E = ∫ E ⋅ d A direction for the area vector d A . These two choices are not independent — they must obey the right-hand rule .
Definition Right-hand rule (loop ↔ surface normal)Curl the fingers of your right hand in the direction you walk around the loop (the direction of the steps d l thumb then points in the direction chosen for the surface normal d A positive side for flux.
Intuition Why this matters for displacement currentThe full Ampère–Maxwell law puts both currents from §2 into the enclosed slot:
∮ B ⋅ d l = μ 0 ( I c + I d ) = μ 0 I c + μ 0 ε 0 d t d Φ E I e n c = I c + I d plus whatever displacement current threads the chosen surface. The sign of the term ε 0 d t d Φ E which way d A . If you reverse your walking direction around the loop, the right-hand rule flips d A both sides of the equation flip sign together — so the physics stays consistent. A growing flux through the thumb side counts as a positive displacement current in the same rotational sense as B
Common mistake Forgetting to tie the two directions togetherIf you pick the loop direction and the normal d A independently , you can get a spurious minus sign and conclude B choose the loop direction first, then let your right thumb dictate d A (or vice versa) — never both by hand.
We met ε 0 μ 0
Intuition Why their product matters (units give a speed)μ 0 ε 0 1 speed of light c Speed of Light . Check the units: F m − 1 × H m − 1 s 2 m − 2 m s − 1 speed . That is the punchline the whole topic builds toward.
σ Surface charge density = charge spread per unit area on a surface:
σ = A Q Q A
Used in Gauss's Law to get the clean result that between capacitor plates E = σ / ε 0
Field = arrows at every point
Electric field E and magnetic field B
Conduction current Ic = dQ/dt
Enclosed current I enc threading a surface
Area vector and dot product
Electric flux PhiE = integral E dot dA
Line integral of B around a loop
Displacement current Id = eps0 dPhiE/dt
Right-hand rule ties loop to normal
Ampere-Maxwell law fixes paradox
Test yourself — reveal only after answering.
What does the little arrow over E It marks a vector — a quantity with both size and direction.
In one phrase, what is a field ? A rule giving an arrow (value + direction) at every point in space .
Write current as a rate of change of charge. I = d t d Q
What does d t d ( something ) "How fast is that something changing right now?" — the rate of change.
What is conduction current I c Real charge flowing through a wire, I c = d Q / d t
What does the subscript e n c I e n c I c I d "Enclosed by the loop" — the total current threading the chosen surface, I e n c = I c + I d I c I d kind , I e n c role .
Which direction does an area vector A Perpendicular to the surface, with length equal to the area.
Why use the dot product E ⋅ A It keeps only the part of
E pointing
through the surface; a field sliding along it (
9 0 ∘ ) contributes nothing since
cos 9 0 ∘ = 0 .
What does electric flux Φ E How many electric-field arrows pierce a given surface.
Define displacement current I d I d = ε 0 d t d Φ E
What does the circle on ∮ The sum runs all the way around a closed loop , back to the start.
What does ∮ B ⋅ d l How much the magnetic field circulates (swirls) around the loop.
How are the loop direction and the surface normal d A By the
right-hand rule : curl right-hand fingers along the walk, thumb gives
d A .
Why does this rule matter for the displacement-current term? It fixes the sign of ε 0 d Φ E / d t
What is ε 0 Permittivity of free space, ≈ 8.85 × 1 0 − 12 F m − 1
What is μ 0 Permeability of free space, 4 π × 1 0 − 7 H m − 1 = N A − 2
Write the full Ampère–Maxwell law with both currents named. ∮ B ⋅ d l = μ 0 ( I c + I d ) = μ 0 I c + μ 0 ε 0 d Φ E / d t .
What famous quantity is 1/ μ 0 ε 0 The speed of light c F m − 1 ⋅ H m − 1 s 2 m − 2 m s − 1
Between capacitor plates, what is E σ 
