Olivier FRUCHART Univ. Grenoble Alpes / CEA / CNRS, SPINTEC, France - PowerPoint PPT Presentation

Olivier FRUCHART Univ. Grenoble Alpes / CEA / CNRS, SPINTEC, France

Dear Institute, I've always had a fascination with electromagnetism, and have pondered the theories of gravity. One thing I've come across in preliminary research is that the current theories largely fail to include human element in, as if we're just baseless objects trapped here without a role in the ultimate reason. (...) Humans are magnets, too, as we possess iron. (…) If you take two magnets, they stick together when proper polars are placed near each other. What causes humans to act as the 2nd magnet in gravity is the iron found in humans. Earth, obviously the big magnet with the most iron, is able to control humans, the far smaller magnet with less iron. (…) Ultimately there is one controlling magnet for the entire universe somewhere in space holding it all together, like Galileo said. Calculations of Earth's maximum gravitation pull could be made by testing individual boosters on humans and converting the thrust needed into some kind of formula which returns Earth's magnetic energical pull . (…) While it doesn't conclude why other things on Earth are in the same situation as us, it is also based on magnetism and humans have to have their own role in the matter. Further research into it needs to be done as these are very preliminary original thoughts. Regards, XXX YYY. Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

What is a quantity? What is a unit ? Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Quantity Units 𝐰 = 𝜀 ℓ /𝜀𝑢 Example: speed Why? dim(𝐰) = L ∙ T −1 Dimension: Provide a measure Universality: share with others Possible formalism: 𝑌 = 𝑌 𝛽 𝑌 𝛽 𝑀 SI = meter = 100 𝑀 cgs 𝑀 = 50 𝑀 SI = 5000 𝑀 cgs SPEED Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Facts: interaction between charges Modeling by the Physicist 𝐆 1→2 = q 2 𝐅 1→2 𝐅 1→2 Electric field 𝑟 1 𝑟 2 𝐆 1→2 = 2 𝐯 12 Charges are scalar sources of electric field 4𝜌𝜗 0 𝑠 12 +𝑟 2 +𝑟 1 𝑟 𝐅(𝐬) = 4𝜌𝜗 0 𝑠 2 𝐯 +𝑟 1 − 𝑟 2 Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Microscopic level: Maxwell equation Macroscopic level: Gauss theroem 𝛂 ⋅ 𝐅 = 𝜍 Ostogradski theorem 𝜗 0 𝒲 𝛂 ⋅ 𝐅 d𝒲 =װ 𝜖𝒲 𝐅 ⋅ 𝐨 d𝒯 ׮ 𝜍 = 𝜀𝑅 Volume density 𝜀𝒲 of electric charge 𝑅 𝜍 𝜗 0 = ׮ 𝜗 0 d𝒲 =װ 𝜖𝒲 𝐅 ⋅ 𝐨 d𝒯 𝒲 Link 𝛂 ⋅ 𝐅 = 𝜖𝐹 𝑦 𝜖𝑦 + … = 𝐹 𝑦 𝑦 + 𝜀𝑦 − 𝐹 𝑦 𝑦 + ⋯ 𝜀𝑦 Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Century-old facts Œrsted experiment in 1820 Magnetic materials (rocks) Light-struck Magnetite Magnetic field of the earth Bir Birth of of ele lect ctromagneti tism Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Facts: interaction between charge currents Modeling by the Physicist Magnetic induction field: Biot & Savart law 𝐽 1 𝐽 2 𝜀 ℓ 2 × (𝜀 ℓ 1 × 𝐯 12 ) +𝐽 𝜀𝐆 1→2 = 𝜈 0 2 4𝜌𝑠 𝜀𝐂 = 𝜈 0 𝐽𝜀 ℓ × 𝐯 12 4𝜌𝑠 2 +𝐽 1 Retrieve the force (Laplace) −𝐽 2 𝜀𝐆 2 = 𝐽 2 𝜀 ℓ × 𝐂(𝐬 2 ) Note: former definition of the Ampère: 𝐆 = 𝑟 𝐰 × 𝐂 The force between two infinitely wires 1m apart with current 1A is 2 × 10 −7 N/m Magnetic in induction fie ield ld defi fined ed through Lo Lorentz Force Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Microscopic level: Maxwell equation Macroscopic level: Ampere theorem 𝛂 × 𝐂 = 𝜈 0 𝐤 Stokes theorem j: Volume density of current (A/m 2 ) 𝝐𝒯 𝐂 ⋅ d ℓ 𝛂 × 𝐂 ⋅ 𝐨 d𝒯 = ׯ ׭ 𝒯 J J is the vectorial source of curl of B 𝝐𝒯 𝐂 ⋅ d ℓ 𝐽 = 𝜈 0 ׭ 𝒯 (𝐤 ⋅ 𝐨) d𝒯 = ׯ 𝐕𝐨𝐣𝐮 𝐠𝐩𝐬 𝐂: tesla (T) Link … … … … 𝛂 × 𝐂 = = 𝜖𝐶 𝑧 𝜖𝑦 − 𝜖𝐶 𝑦 𝐶 𝑧 𝑦 + 𝜀𝑦 − 𝐶 𝑧 𝑦 − 𝐶 𝑦 𝑧 + 𝜀𝑧 − 𝐶 𝑦 (𝑧) 𝜖𝑧 𝜀𝑦 𝜖𝑧 Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

𝛂 ⋅ 𝐅 = 𝜍 Gauss theorem 𝜗 0 𝛂 × 𝐅 = − 𝜖𝐂 Faraday law of induction 𝜖𝑢 𝜖𝐅 𝛂 × 𝐂 = 𝜈 0 𝐤 + 𝜗 0 Ampère theorem 𝜖𝑢 𝛂 ⋅ 𝐂 = 0 B is divergence free (no magnetic poles) Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Biot and Savart The magnetic point dipole 𝜀𝐂 = 𝜈 0 𝐽𝜀 ℓ × 𝐯 4𝜌𝑠 2 Simple loop Note: 1/ r 2 dec 𝐕𝐨𝐣𝐮: A ⋅ m 2 𝛎 = 𝐽𝒯 𝐨 Not ecay General definition Ampere theorem and Œrsted field 𝛎 = 1 𝐽 2 ම 𝐬 × 𝐤 𝐬 d𝒲 𝒲 𝐶 𝜄 = 𝜈 0 𝐽 2𝜌𝑠 𝜈 0 3 ote: 1/ r 3 dec Not ecay 𝐂 = 𝑠 2 𝛎 ⋅ 𝐬 𝐬 − 𝛎 4𝜌𝑠 3 𝜈 0 Not ote: 1/ r dec ecay 𝐂 = 4𝜌𝑠 3 2𝜈 cos 𝜄 𝐯 𝑠 + 𝜈 sin 𝜄 𝐯 𝜄 Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Energy Torque (𝐾) ℰ = −𝛎 ⋅ 𝐂 Zeeman energy 𝚫 = ර 𝐬 × 𝐽 d ℓ × 𝐂 = 𝛎 × 𝐂 Demonstration Work to compensate Lenz law Inducing precession of dipole around the field during rise of B It is energy-conservative, as expected from Laplace Integrate torque from Laplace (Lorentz) force force while flipping dipole in B Force 𝐆 = 𝛎 ⋅ (𝛂𝐂) Valid only for fixed dipole No force in uniform magnetic induction field Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Energy Examples ℰ = − 𝜈 0 3 ℰ = +2 𝜈 0 𝜈 1 𝜈 2 𝑠 2 𝛎 1 ⋅ 𝐬 𝛎 2 ⋅ 𝐬 − 𝛎 𝟐 ⋅ 𝛎 2 4𝜌𝑠 3 4𝜌𝑠 3 ℰ = + 𝜈 0 𝜈 1 𝜈 2 The dipole-dipole interaction is anisotropic 4𝜌𝑠 3 ℰ = 0 ℰ = − 𝜈 0 𝜈 1 𝜈 2 4𝜌𝑠 3 ℰ = −2 𝜈 0 𝜈 1 𝜈 2 4𝜌𝑠 3 Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Definition Equivalence with surface currents Volume density of magnetic point dipoles 𝐍 = 𝜀𝛎 A/ m 𝜀𝒲 Total magnetic moment of a body A ⋅ m 2 𝓝 = න 𝐍 d𝒲 𝒲 Appli lies es to: o: fer erromagnets, paramagnets, dia iamagnets etc. c. Must be be defin ined ed at at a len ength scale Name: Amperian description of magnetism much la larger th than atom oms Surface current equals magnetization A/ m Is the basis for th Is the micromagnetic th theo eory Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Back to Maxwell equations The magnetic field H Disregard fast time dependence: 𝐂 𝛂 × − 𝐍 = 𝐤 c magnetostatics One has: 𝜈 0 𝜖𝐅 𝛂 × 𝐂 = 𝜈 0 𝐤 + 𝜗 0 𝜖𝑢 𝐈 = 𝐂 A/ m − 𝐍 By definition: 𝐤 c 𝜈 0 Consider separately real charge current, 𝐤 m from fictitious currents of magnetic dipoles 𝛂 × 𝐈 = 𝐤 c 𝛂 × 𝐂 = 𝜈 0 𝐤 c + 𝐤 m B versus H : definition of the system A/m 2 𝛂 × 𝐍 = 𝐤 m One can show: 𝜀𝒲 M: local (infinitesimal) part in of the 𝐍 × 𝐨 = 𝐤 m,s A/m system defined when considering a magnetic material H: The remaining of B coming from outside 𝜈 0 𝐈 Outs tsid ide e matter, and c coin oincid ide and 𝐂 𝜀𝒲 , liable to interact with the system have e exactly th the e same mea eaning. Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Derive the dipolar field The dipolar field H d By definition: the contribution to H 𝛂 ⋅ 𝐂 = 𝟏 𝛂 ⋅ 𝐈 d = −𝛂 ⋅ 𝐍 Maxwell equation → not related to free currents (possible to split as Maxwell equations are 𝛂 ⋅ 𝐧 𝐬 ′ (𝐬 − 𝐬 ′ ) linear) 𝐈 d 𝐬 = −𝑁 s ම d𝒲′ 4𝜌 𝐬 − 𝐬 ′ 3 𝛂 × 𝐈 d = 0 𝐈 d = −𝛼𝜚 d 𝒲 ′ 𝐈 = 𝐈 d + 𝐈 𝐛𝐪𝐪 To lift the singularity that may arise at boundaries, a volume integration around the boundaries yields: Analogy with electrostatics 𝐈 d 𝐬 = ම 𝜍 𝐬 ′ 4𝜌 𝐬 − 𝐬 ′ 3 d𝒲 ′ + ඾ 𝜏 𝐬 ′ 𝐬 − 𝐬 ′ 𝐬 − 𝐬 ′ 4𝜌 𝐬 − 𝐬 ′ 3 d𝒯 ′ 𝛂 × 𝐅 = 0 𝐅 = −𝛼𝜚 𝜍(r)= − 𝑁 s 𝛂 ⋅ 𝐧(𝐬) 𝜏(r)=𝑁 s 𝐧 𝐬 ⋅ 𝐨(𝐬) Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Example Permanent magnet (uniformly-magnetized) Vocabulary Generic names Magnetostatic field Dipolar field Inside material Demagnetizing field Oustide material Stray field Surface charges Dipolar field 𝐈 d 𝐬 = ඾ 𝜏 𝐬 ′ 𝐬 − 𝐬 ′ 4𝜌 𝐬 − 𝐬 ′ 3 d𝒯 ′ 𝜏(r)=𝑁 s 𝐧 𝐬 ⋅ 𝐨(𝐬) Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic

Amperian Coulombian Pseudo-charges source of Hd Fictitious currents source of B 𝛂 × 𝐈 = 𝟏 𝛂 ⋅ 𝐂 = 𝟏 No closed No magnetic lines monopole Δ𝐶 ⊥ = 0 Δ𝐼 ∥ = 0 Δ𝐂 = 𝜈 0 𝐤 × 𝐨 Δ𝐈 ⋅ 𝐨 = 𝜏 From: M. Coey’s book Olivier FRUCHART – Fields, moments, units ESM2019, Brno, Czech Republic