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Content

Magnetic Field of a Straight Wire
 Magnetic Field along Axis of Circular Loop of Wire
 Permanent Magnet
 Source and Reference

Magnetic Field of a Straight Wire

The magnetic field of a long straight wire is

𝑟=<𝑥,0,−𝑧>⇒{|𝑟|²=𝑥²+𝑧²𝑟=𝑟|𝑟|=<𝑥,0,−𝑧>𝑥²+𝑧²

∆𝑙=∆𝑧=<0,0,∆𝑧>⇒∆𝑧×𝑟=1|𝑟|(∆𝑧×𝑟)=1|𝑟|𝑥 𝑦 𝑧0 0 ∆𝑧𝑥 0 −𝑧=0𝑥+𝑥∆𝑧𝑦+0𝑧|𝑟|=𝑥∆𝑧𝑥²+𝑧²𝑦

∆𝐵=𝜇₀4𝜋𝐼∆𝑙×𝑟|𝑟|²=𝜇₀4𝜋𝐼∆𝑧×𝑟𝑥²+𝑧²=𝜇₀𝐼4𝜋𝑦𝑥∆𝑧(𝑥²+𝑧²)^3/2

𝐵=∫∆𝐵=𝜇₀𝐼4𝜋𝑦𝐿/2∫−𝐿/2𝑥(𝑥²+𝑧²)^3/2𝑑𝑧

    |𝐵_𝑦|=𝜇₀4𝜋𝐼𝐿𝑥𝑥²+(𝐿/2)²

By symmetry, the magnetic field of a long straight wire can be expressed in cylindrical coordinates with 𝑥=𝑅 as following 𝐵=𝜇₀4𝜋𝐼𝐿𝑅𝑅²+(𝐿/2)²𝜃 When the length of wire tend to infinite long the measured point is very close to the wire, 𝑅≪𝐿

𝑅≪𝐿⇒𝑅²+(𝐿/2)²≈𝐿/2

         𝐵=𝜇₀4𝜋𝐼𝐿𝑅(𝐿/2)𝜃=𝜇₀4𝜋2𝐼𝑅𝜃

Magnetic Field along Axis of Circular Loop of Wire

By symmetry, the magnetic field is reduced to 𝑧-component. and the magnetic field of one element is

∆𝐵_𝑧=𝜇₀4𝜋𝐼𝑅²∆𝜃(𝑅²+𝑧²)^3/2

𝐵_𝑧=∫∆𝐵_𝑧=2𝜋∫0𝜇₀4𝜋𝐼𝑅²∆𝜃(𝑅²+𝑧²)^3/2=𝜇₀4𝜋𝐼𝑅²2𝜋(𝑅²+𝑧²)^3/2=𝑇∙𝑚𝐴(𝑚²∙𝐴)(𝑚²)^3/2=[𝑇]

For 𝑧 far from the loop, 𝑧≫𝑅 (𝑧²+𝑅²)^3/2=(𝑧²)^3/2=𝑧³ Therefore the magnetic field along axis far from loop

𝐵_axis≈𝜇₀4𝜋2𝜋𝑅²𝐼𝑧³

        Let Magnetic dipole moment 𝜇=𝐼𝐴; 𝐴=𝜋𝑅²⇒𝐵_axis≈𝜇₀4𝜋2𝜇𝑟³

Similar to electric field Let electric dipole moment 𝑝=𝑞𝑠⇒𝐸_axis≈14𝜋𝜀₀2𝑝𝑟³

Permanent Magnet

Current loop induces electromagnet, while electrons orbit atoms are atomic scale electomagnet. Electron orbiting atom in a p-orbital. When all the atomic magnet align, the material is a ferromagnet. The temperature at which atomic magnets align is called the curie temperature and is arel phase transition of the electrons inside a material, e.g. about 1000K. Electron orbits atom and electron also spins. Both act like electromagnets and both contribut to permanent magnet.
Some common ferromagnetic materials: iron, cobalt, nickel. The alignment of atomic dipole moments in ferromagnetic materials is along one direction to produce a strong magnetic as permanent magnet. The magnetic domains of ferromagnetic materials can also aligned by induced core. However, the magnetic field can also be demagnetized by hitting or heating.