Electromagnetic Waves: Electric Current and Current Density

Electromagnetic Waves, Antennas, and RF: Electric Current and Current Density

EM05–EM07 treated Electric Charge primarily as a static source. The next step is to let charge move.

Electric current measures how rapidly charge crosses an oriented surface. Current density describes how that charge flow is distributed in space and in direction. These ideas are essential because moving charge produces magnetic effects, and time-varying currents are among the physical sources of electromagnetic radiation and antenna fields [1, 2, 3, 5].

The central distinction is

|-------| I = dq-| -----dt--

(1)

for current through a chosen surface, whereas

|--------------------| -J-=-current-density-|

(2)

is a vector field defined throughout a region of space.

The two are connected by

|----∫---------| |I = J ⋅ dA.| | S | ---------------

(3)

This equation should look familiar. It has the same geometric structure as the electric-flux integral introduced in EM06.

1 Electric current as charge flow rate

Suppose an oriented surface is placed across a conducting wire. Let dq be the net charge that crosses the surface in time dt. The instantaneous current is

|--------| | dq | |I = dt-.| ---------

(4)

The SI unit of current is the ampere:

|------------| -1A-=--1C/s.--

(5)

If a constant current I flows for a time interval Δt, the amount of charge transported is

|----------| Δq--=-I-Δt.-

(6)

Current is therefore not an amount of charge. It is a rate of charge transport.

Figure. Electric current measures the net charge crossing an oriented surface per unit time. The arrow labeled I indicates the conventional-current direction.

2 Conventional current direction

By convention, current direction is defined as the direction in which positive charge would move.

If positive charge carriers move toward +x, the conventional current points toward +x.

In an ordinary metal, the mobile carriers are electrons. Because electrons have negative charge, their drift direction is opposite the conventional-current direction.

This convention predates the discovery of the electron, but it remains standard throughout circuit theory and Electromagnetism.

2.1 Current is signed

Once a positive surface orientation has been chosen, current can be positive or negative. A net positive charge flow in the chosen positive direction gives positive current. Reversing the orientation reverses the sign assigned to the same physical flow.

3 From current to current density

Current tells us the total charge flow through a surface, but not how that flow is distributed across the surface.

Imagine a wide Conductor. The charge flow might be nearly uniform, or it might be concentrated near one side. To describe the local flow we introduce the current density

|------| J-(r,t).-

(7)

Current density is a vector field. Its direction gives the local conventional-current direction, and its magnitude gives current per unit area measured perpendicular to the flow.

Its SI unit is

|----------2-| -[J-] =-A/m--.-

(8)

4 Current as flux of current density

Consider a small oriented surface element

dA = ˆn dA.

(9)

The infinitesimal current through that patch is

|------------| -dI-=-J-⋅ dA.-

(10)

Using the dot product,

dI = J dA cos 𝜃,

(11)

where 𝜃 is the angle between J and the surface normal.

Adding all the surface patches gives

|----∫---------| |I = J ⋅ dA.| ------S--------|

(12)

This is the current-density analogue of a flux integral.

Figure. Only the component of J normal to the surface contributes to current through the surface. The angle is measured between J and the area normal n.

5 Uniform current density

If J is uniform over a flat area A and perpendicular to the surface, then

J ⋅ dA = J dA.

(13)

Therefore,

I	= ∫ _SJ dA	(14)
	= JA.	(15)

Hence,

|--------| -I-=-J-A-|

(16)

and

|--------| |J = I-.| ------A--|

(17)

This familiar formula is a special case of the surface integral. It is not valid when J varies appreciably across the cross section unless J is interpreted as an average.

6 Current density from moving charge density

Suppose a continuous volume charge density ρ moves with velocity v.

In a short time dt, charge moving normally through a surface sweeps out a volume proportional to v dt. The amount of charge crossing per unit area per unit time is therefore proportional to ρv.

The vector relation is

|--------| -J-=-ρv.-|

(18)

This formula is especially useful when the moving charge can be modeled as a continuous fluid.

The units confirm the interpretation:

-C-m- = -C--= A--. m3 s m2s m2

(19)

7 Microscopic carrier form: J = nqv_d

Suppose there are n mobile charge carriers per unit volume. If each carrier has charge q, the mobile volume charge density is

ρ = nq.

(20)

If their average drift velocity is v_d, then

|----------| |J = nqvd. | -----------

(21)

For positive carriers, J points in the same direction as v_d.

For electrons,

q = − e,

(22)

|------------| |J = − nevd. | -------------

(23)

Thus the conventional current density points opposite the electron drift velocity.

Figure. In a metal, electrons drift opposite to the conventional-current direction because the mobile carriers have negative charge.

8 Drift velocity is usually slow

The electrical signal in a conductor can propagate rapidly even though the average drift speed of individual charge carriers is often quite small.

For a wire carrying a uniform current,

I = JA = nqvdA

(24)

in magnitude. Therefore,

|------------| | I | |vd = ------.| ------|q|nA---

(25)

Because conductors contain an enormous number of mobile electrons per unit volume, modest currents generally require only small average drift speeds.

This distinction becomes important later in RF. Electromagnetic disturbances and fields can propagate through a structure far faster than individual electrons drift along the conductor.

9 Example 1: charge transported by a steady current

A steady current of

I = 2.5A

(26)

flows for

Δt = 8.0s.

(27)

The transported charge is

Δq	= IΔt	(28)
	= (2.5)(8.0)	(29)
	= 20 C.	(30)

Thus,

|-----------| Δq--=-20-C.--

(31)

10 Example 2: uniform current density in a wire

A wire carries

I = 3.0A

(32)

through a cross-sectional area

−6 2 A = 1.5 × 10 m .

(33)

Assuming uniform current density perpendicular to the cross section,

J	=	(34)
	=	(35)
	= 2.0 × 10⁶ A/m².	(36)

Therefore,

|--------------------| | 6 2 | -J-=-2.0-×-10-A/m---.-

(37)

11 Example 3: current through a tilted surface

A uniform current density has magnitude

2 J = 5.0A/m .

(38)

It crosses a flat area

A = 0.40 m2

(39)

whose normal makes an angle

∘ 𝜃 = 60

(40)

with J.

Then

I	= JA cos 𝜃	(41)
	= (5.0)(0.40) cos 60^∘	(42)
	= 1.0 A.	(43)

|----------| -I-=-1.0A.--

(44)

12 Example 4: electron drift speed

A metallic wire carries

I = 1.0A.

(45)

Suppose

28 −3 −6 2 n = 8.5 × 10 m , A = 1.0 × 10 m ,

(46)

and each carrier has charge magnitude

−19 e = 1.602 × 10 C.

(47)

The drift-speed magnitude is

v_d	=	(48)
	=	(49)
	≈ 7.34 × 10⁻⁵ m/s.	(50)

Thus,

|--------------−5------| -vd-≈-7.34-×-10---m/s.-|

(51)

This is less than one tenth of a millimeter per second.

13 Charge conservation for a fixed volume

Charge cannot simply disappear from a region without crossing its boundary or being balanced by corresponding charge transport.

Consider a fixed volume V bounded by a closed surface S. The total charge inside is

|---------∫------------| |QV (t) = ρ(r,t)dV. | -----------V-----------|

(52)

If positive current flows outward through the boundary, the charge remaining inside decreases. Therefore,

|-d-∫------------∮---------| |-- ρdV = − J ⋅ dA. | -dt--V------------S--------|

(53)

The minus sign is essential: positive outward current corresponds to decreasing enclosed charge.

Figure. Charge conservation for a fixed control volume. Net outward current through the boundary reduces the charge stored inside the volume.

14 From integral conservation to the continuity equation

Use the divergence theorem:

∮ ∫ J ⋅ dA = ∇ ⋅ J dV. S V

(54)

Also, for a fixed volume,

d ∫ ∫ ∂ρ -- ρ dV = ---dV. dt V V ∂t

(55)

Therefore,

∫ ∫ ∂ρ-dV = − ∇ ⋅ JdV. V ∂t V

(56)

Move everything to one side:

∫ ( ∂ρ ) ---+ ∇ ⋅ J dV = 0. V ∂t

(57)

Because this must hold for every sufficiently small volume,

|----------------| |∂ρ- | |∂t + ∇ ⋅ J = 0.| -----------------

(58)

This is the continuity equation for electric charge.

15 Physical meaning of the continuity equation

The continuity equation says

|---------------------------------------------------------------| local-charge--accumulation-=--− local-outward--current-divergence.|

(59)

∇ ⋅ J > 0,

(60)

more current leaves a tiny region than enters it, so

∂ρ ---< 0. ∂t

(61)

The local charge density decreases.

∇ ⋅ J < 0,

(62)

more current enters than leaves, so charge accumulates locally.

This gives a physical interpretation to the divergence operator introduced in EM03.

16 Steady current

For a truly steady charge-flow pattern,

∂ρ- ∂t = 0.

(63)

The continuity equation then gives

|----------| -∇-⋅ J-=-0.|

(64)

A steady current density therefore has no local accumulation or depletion of charge in the region being modeled.

This does not mean that J must be spatially uniform. It means only that its divergence is zero.

17 Example 5: current density from moving volume charge

Suppose

−6 3 ρ = 4.0 × 10 C/m

(65)

moves with velocity

v = 3.0ˆx m/s.

(66)

Then

J	= ρv	(67)
	= (4.0 × 10⁻⁶)(3.0)x	(68)
	= 1.2 × 10⁻⁵x A/m².	(69)

Therefore,

|----------------------| J = 1.2 × 10 −5ˆx A/m2.| ------------------------

(70)

18 Example 6: nonuniform current density through a disk

Suppose a circular cross section of radius R carries an axial current density

( 2 ) J(s) = J 1 − s-- ˆz, 0 R2

(71)

where s is radial distance from the axis.

For a cross section normal to z,

dA = sds dϕ.

(72)

Thus,

I	= ∫ ₀^2π ∫ ₀^RJ₀sdsdϕ	(73)
	= 2πJ₀₀^R	(74)
	= 2πJ₀	(75)
	= .	(76)

Therefore,

|------------| | 2 | |I = πJ0R---.| --------2----

(77)

This example shows why the general surface integral is needed when current density is not uniform.

19 Example 7: continuity equation and local charge loss

Let

J = αx ˆx,

(78)

where α is constant.

The divergence is

∂(αx ) ∇ ⋅ J = ------ = α. ∂x

(79)

The continuity equation gives

∂ρ-= − ∇ ⋅ J = − α. ∂t

(80)

Hence,

|----------| |∂ρ | |---= − α.| -∂t---------

(81)

The positive divergence means net current leaves a small region, so the local charge density decreases with time.

20 Current density and the antenna path ahead

Current density is not merely a circuit quantity. It is one of the central source fields of classical electromagnetism.

Later articles will introduce magnetic fields produced by currents and then Maxwell’s equations. In those equations, the source term J(r,t) describes electric current density distributed through space.

For antennas, the important source is generally time-varying current. A wire antenna, for example, supports a current distribution that varies with position and time. The spatial and temporal structure of that current determines the electromagnetic fields radiated into space.

A simplified harmonic current can be written schematically as

J(r,t) = J0(r)cos(ωt + ϕ).

(82)

The detailed connection between such currents and radiation will be developed only after the magnetic field, Faraday’s law, and the Maxwell-Ampere equation are established.

21 Current is not the same as current density

These quantities must remain distinct:

I versus J.

(83)

Current I is a scalar rate through a selected surface and has units of amperes.

Current density J is a vector field and has units of amperes per square meter.

They are related by

|----∫---------| |I = J ⋅ dA.| | S | ---------------

(84)

Knowing I alone does not determine the full spatial distribution of J.

22 Common mistakes

Confusing charge with current. Charge is measured in coulombs; current is charge per unit time.
Confusing I with J. Current is a scalar through a chosen surface; current density is a vector field.
Ignoring surface orientation. The sign of ∫ J ⋅ dA depends on the chosen normal direction.
Using I = JA for a nonuniform J. The general relation is the surface integral.
Assuming electrons move in the current direction. In metals, electron drift is opposite conventional current.
Assuming rapid electrical response means rapid electron drift. Carrier drift can be slow even while electromagnetic disturbances propagate rapidly.
Forgetting the minus sign in charge conservation. Net outward current decreases the charge stored inside a fixed volume.

Summary

The essential results of EM08 are:

Electric current is the rate of charge flow:
$dq I = ---. dt$
Current density is a vector field with units A/m².
Current through a surface is
$∫ I = S J ⋅ dA.$
For uniform perpendicular current density,
$I = JA.$
Moving continuous charge gives
$J = ρv.$
For discrete carrier density n,
$J = nqv . d$
Charge conservation gives
$∫ ∮ -d dt V ρdV = − S J ⋅ dA.$
The local continuity equation is
$∂ρ ---+ ∇ ⋅ J = 0. ∂t$

The next lesson, EM09, introduces magnetic fields and the magnetic force on moving charge, beginning the transition from charge flow to the coupled electric-and-magnetic description required for electromagnetic waves.

References

[1] David J. Griffiths, Introduction to Electrodynamics, 4th ed., Cambridge University Press, 2017.

[2] Samuel J. Ling, Jeff Sanny, and William Moebs, University Physics, Volume 2, OpenStax, 2016, sections on electric current, current density, and charge conservation.

[3] Edward M. Purcell and David J. Morin, Electricity and Magnetism, 3rd ed., Cambridge University Press, 2013.

[4] Richard P. Feynman, Robert B. Leighton, and Matthew Sands, The Feynman Lectures on Physics, Volume II, Addison-Wesley, 1964, chapters on current, charge conservation, and electromagnetic fields.

[5] Massachusetts Institute of Technology, 8.02 Physics II: Electricity and Magnetism, MIT OpenCourseWare, materials on current, current density, magnetic fields, and Maxwell’s equations.

[6] John D. Jackson, Classical Electrodynamics, 3rd ed., Wiley, 1999, sections on charge conservation and current density.

"Electromagnetic Waves: Electric Current and Current Density" is owned by bloftin.
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Electromagnetic Waves, Antennas, and RF: Electric Current and Current Density

1 Electric current as charge flow rate

2 Conventional current direction

2.1 Current is signed

3 From current to current density

4 Current as flux of current density

5 Uniform current density

6 Current density from moving charge density

7 Microscopic carrier form: J = nqvd

8 Drift velocity is usually slow

9 Example 1: charge transported by a steady current

10 Example 2: uniform current density in a wire

11 Example 3: current through a tilted surface

12 Example 4: electron drift speed

13 Charge conservation for a fixed volume

14 From integral conservation to the continuity equation

15 Physical meaning of the continuity equation

16 Steady current

17 Example 5: current density from moving volume charge

18 Example 6: nonuniform current density through a disk

19 Example 7: continuity equation and local charge loss

20 Current density and the antenna path ahead

21 Current is not the same as current density

22 Common mistakes

Summary

References

7 Microscopic carrier form: J = nqv_d