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
|𝐵𝑦|=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 ∆𝐵𝑧=𝜇04𝜋𝐼𝑅2∆𝜃(𝑅2+𝑧2)3/2
𝐵𝑧=∫∆𝐵𝑧=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 𝐵axis≈𝜇04𝜋2𝜋𝑅2𝐼𝑧3
Let Magnetic dipole moment 𝜇=𝐼𝐴; 𝐴=𝜋𝑅2⇒𝐵axis≈𝜇04𝜋2𝜇𝑟3 Similar to electric field Let electric dipole moment 𝑝=𝑞𝑠⇒𝐸axis≈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.

Source and Reference

https://www.youtube.com/watch?v=JnhxIY0mHuw&list=PLZ6kagz8q0bvxaUKCe2RRvU_h7wtNNxxi&index=13