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Wed 21 Sep, 2022 12:35 pm

In this work are investigated details of an elementary electric and magnetic dipole radiation at distances much greater than size of the emitting element. Debatable conclusions are drawn.

Formulas are given in a cylindrical coordinate system (ρ,φ,z)

r^2 = ρ^2 + z^2

Given this, it is possible to write expressions differently for ρ and z

For example, 2 - 3 · ρ^2 / r^2 = 3 · z^2 / r^2 - 1

For all values ∂/∂φ = 0 (cylindrical symmetry)

Time derivatives are denoted by a quote '

**Electric elementary dipole**

Charge oscillates along z-axis near zero point with frequency ω, amplitude of dipole moment is P0.

Dipole moment:

Pz = P0 · cos(ω·t)

Auxiliary functions:

COS = cos(ω·(t - r/c)), SIN = sin(ω·(t - r/c))

Scalar potential:

a = P0 / (4·π·ε0) · z / r^2 · (1 / r · COS - ω/c · SIN)

a' = - P0 / (4·π·ε0) · ω · z / r^2 · (ω/c · COS + 1 / r · SIN)

Vector potential:

Az = - P0 · μ0/(4·π) · ω / r · SIN

Az' = - P0 · μ0/(4·π) · ω^2 / r · COS

div A = ∂Az/∂z = P0 · μ0/(4·π) · ω · z / r^2 · (ω/c · COS + 1 / r · SIN)

a' = - c^2 · div A

Scalar potential gradient:

∂a/∂ρ = P0 / (4·π·ε0) · ρ · z / r^3 · {(ω^2/c^2 - 3 / r^2) · COS + ω/c · 3 / r · SIN}

∂a/∂z = P0 / (4·π·ε0) / r^2 · {1 / r · (ω^2/c^2 · z^2 + 1 - 3 · z^2 / r^2) · COS + ω/c · (3 · z^2 / r^2 - 1) · SIN}

Magnetic induction:

Bφ = - ∂Az/∂ρ = - P0 · μ0/(4·π) · ω · ρ / r^2 · (ω/c · COS + 1 / r · SIN)

Bφ' = - P0 · μ0/(4·π) · ω^2 · ρ / r^2 · (1 / r · COS - ω/c · SIN)

Electric field:

Eρ = - ∂a/∂ρ = - P0 / (4·π·ε0) · ρ · z / r^3 · {(ω^2/c^2 - 3 / r^2) · COS + ω/c · 3 / r · SIN}

Ez = - Az' - ∂a/∂z = P0 / (4·π·ε0) / r · {(ω^2/c^2 · ρ^2 / r^2 - 1 / r^2 + 3 · z^2 / r^4) · COS + ω/c / r · (1 - 3 · z^2 / r^2) · SIN}

Electric field curl:

∂Eρ/∂z - ∂Ez/∂ρ = P0 / (4·π·ε0) · ω^2/c^2 · ρ / r^2 · (1 / r · COS -ω/c · SIN)

Bφ' = - (∂Eρ/∂z - ∂Ez/∂ρ)

as it should be in equations of electromagnetic field.

div E = ∂Eρ/∂ρ + Eρ / ρ + ∂Ez/∂z = 0 (checked)

Magnetic field curl:

Jρ = - 1/μ0 · ∂Bφ/∂z = - P0 / (4·π) · ω · ρ · z / r^3 · {ω/c · 3 / r · COS - (ω^2/c^2 - 3 / r^2) · SIN}

Jz = 1 / μ0 · (∂Bφ/∂ρ + Bφ / ρ) = P0 / (4·π) · ω / r · {ω/c / r · (1 - 3 · z^2 / r^2) · COS - (ω^2/c^2 · ρ^2 / r^2 - 1 / r^2 + 3 · z^2 / r^4) · SIN}

Eρ' = - P0 / (4·π·ε0) · ω · ρ · z / r^3 · {ω/c · 3 / r · COS - (ω^2/c^2 - 3 / r^2) · SIN} = Jρ/ε0

Ez' = P0 / (4·π·ε0) · ω / r · {ω/c / r · (1 - 3 · z^2 / r^2) · COS - (ω^2/c^2 · ρ^2 / r^2 - 1 / r^2 + 3 · z^2 / r^4) · SIN} = Jz/ε0

as it should be in equations of electromagnetic field.

**Magnetic dipole**

An annular current with small radius R changes direction according to periodic law.

Magnetic moment is directed along z-axis:

Mz = M0 · cos(ω·t), где M0 = π · R^2 · I0, I0 is current amplitude.

Auxiliary functions:

COS = cos(ω·(t - r/c)), SIN = sin(ω·(t - r/c))

Vector potential:

Aφ = M0 · μ0/(4·π) · ρ / r^2 · (1 / r · COS - ω/c · SIN)

Electric field:

Eφ = - Aφ' = M0 · μ0/(4·π) · ω · ρ / r^2 · (ω/c · COS + 1 / r · SIN)

Eφ' = M0 · μ0/(4·π) · ω^2 · ρ / r^2 · (1 / r · COS - ω/c · SIN)

Magnetic induction:

Bρ = - ∂Aφ/∂z = - M0 · μ0/(4·π) · ρ · z / r^3 · {(ω^2/c^2 - 3 / r^2) · COS + ω/c · 3 / r · SIN}

Bz = ∂Aφ/∂ρ + Aφ / ρ = M0 · μ0/(4·π) / r^2 · {(ω^2/c^2 · ρ^2 / r + 2 / r - 3 · ρ^2 / r^3) · COS - ω/c · (2 - 3 ·ρ^2 / r^2) · SIN}

Bρ' = - M0 · μ0/(4·π) · ω · ρ · z / r^3 · {ω/c · 3 / r · COS - (ω^2/c^2 - 3 / r^2) · SIN} = - (- ∂Eφ/∂z)

Bz' = - M0 · μ0/(4·π) · ω / r^2 · {ω/c · (2 - 3 · ρ^2 / r^2) · COS + (ω^2/c^2 · ρ^2 / r + 2 / r - 3 · ρ^2 / r^3) · SIN} = - (∂Eφ/∂ρ + Eφ / ρ)

as it should be in equations of electromagnetic field.

Magnetic field curl:

Jφ = 1/μ0 · (∂Bρ/∂z - ∂Bz/∂p) = M0 · μ0/(4·π) · ω^2/c^2 · ρ / r^2 · {1 / r · COS - ω/c · SIN}

Eφ' = Jφ/ε0 (checked)

as it should be in equations of electromagnetic field.

**Conclusions**

Although divergence of electric field div(E) is zero everywhere (charge density is zero), scalar potential is urgently needed to describe radiation of electric dipole. To express time derivative a' is required vector potential A. At long distances, there is no question of lagging potentials of forcibly oscillating system, waves must propagate "by themselves" in wave zone. It begs the conclusion that potentials are objective physical reality, fundamental fields in vacuum, and are not mathematical abstractions. To describe dipole radiation, three fundamental fields are sufficient:

a' = - c^2 · div A

A' = - E - grad a

E' = c^2 · rot rot A

At the same time, Laplacian div grad (a) is fundamentally different from local charge density ε0 · div E, these are different quantities. Laplacian of scalar potential can be locally not zero in electric dipole radiation, unlike divergence of electric field. Formally, both of these quantities are "conserved", since it is possible to express derivatives in time as minus divergence of some known "flow" or current. But with respect to electric dipole, laplasian of scalar potential is preserved only globally, when positive density is emitted in one direction along z-axis, in opposite direction the same modulo negative goes. It cannot be said that scalar potential has significant value only in near zone of forced generation and lagging potentials. In far wave zone, its intensity, like time derivative, decreases proportionally to 1 / r along z-axis, the same applies to its gradient in some directions (ρ · z / r^3).

Electric and magnetic field decrease on average with distance as 1 / r, respectively, energy density decreases as 1 / r^2. That is, integral of energy density throughout space is infinite, and elementary dipoles cannot be used as basis for representing field objects with finite energy. The more time emitter works, more energy it loses with waves, without restrictions on final value.

"Electric and magnetic field decrease on average with distance as 1 / r, respectively..."

Don't the field due to a point charge or bar magnet fall off as the inverse square of distance?

@Brandon9000,

I wished to emphasize the difference between forced wave generation processes and independent field propagation. Historically physicists first used currents to generate magnetic field or motion of magnet to generate electric field, and new field depended on changes over time of old. Then they understood that fields can exist independently, and there are different cause and effect sequence: change with time of any fundamental field (first derivative over time) is the consequence and must be in one part of equation, and the cause is some spatial state of the same field and other fields and must be in other part of equation, without time derivatives. Understanding and distinguishing this, we will not make mistakes in the calculations.