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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

image The magnetic field of a long straight wire is π‘Ÿ=<π‘₯,0,βˆ’π‘§>β‡’{|π‘Ÿ|2=π‘₯2+𝑧2π‘Ÿ=π‘Ÿ|π‘Ÿ|=<π‘₯,0,βˆ’π‘§>π‘₯2+𝑧2
βˆ†π‘™=βˆ†π‘§=<0,0,βˆ†π‘§>β‡’βˆ†π‘§Γ—π‘Ÿ=1|π‘Ÿ|(βˆ†π‘§Γ—π‘Ÿ)=1|π‘Ÿ|π‘₯ 𝑦 𝑧0 0β€‚βˆ†π‘§π‘₯ 0β€‚βˆ’π‘§=0π‘₯+π‘₯βˆ†π‘§π‘¦+0𝑧|π‘Ÿ|=π‘₯βˆ†π‘§π‘₯2+𝑧2𝑦
βˆ†π΅=πœ‡04πœ‹πΌβˆ†π‘™Γ—π‘Ÿ|π‘Ÿ|2=πœ‡04πœ‹πΌβˆ†π‘§Γ—π‘Ÿπ‘₯2+𝑧2=πœ‡0𝐼4πœ‹π‘¦π‘₯βˆ†π‘§(π‘₯2+𝑧2)3/2
𝐡=βˆ«βˆ†π΅=πœ‡0𝐼4πœ‹π‘¦πΏ/2βˆ«βˆ’πΏ/2π‘₯(π‘₯2+𝑧2)3/2𝑑𝑧
|𝐡𝑦|=πœ‡04πœ‹πΌπΏπ‘₯π‘₯2+(𝐿/2)2
By symmetry, the magnetic field of a long straight wire can be expressed in cylindrical coordinates with π‘₯=𝑅 as following 𝐡=πœ‡04πœ‹πΌπΏπ‘…π‘…2+(𝐿/2)2πœƒ When the length of wire tend to infinite long the measured point is very close to the wire, 𝑅β‰ͺ𝐿 𝑅β‰ͺ𝐿⇒𝑅2+(𝐿/2)2β‰ˆπΏ/2
𝐡=πœ‡04πœ‹πΌπΏπ‘…(𝐿/2)πœƒ=πœ‡04πœ‹2πΌπ‘…πœƒ

Magnetic Field along Axis of Circular Loop of Wire

image By symmetry, the magnetic field is reduced to 𝑧-component. and the magnetic field of one element is βˆ†π΅π‘§=πœ‡04πœ‹πΌπ‘…2βˆ†πœƒ(𝑅2+𝑧2)3/2
𝐡𝑧=βˆ«βˆ†π΅π‘§=2πœ‹βˆ«0πœ‡04πœ‹πΌπ‘…2βˆ†πœƒ(𝑅2+𝑧2)3/2=πœ‡04πœ‹πΌπ‘…22πœ‹(𝑅2+𝑧2)3/2=π‘‡βˆ™π‘šπ΄(π‘š2βˆ™π΄)(π‘š2)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 𝐡axisβ‰ˆπœ‡04πœ‹2πœ‹π‘…2𝐼𝑧3
Let Magnetic dipole moment πœ‡=𝐼𝐴; 𝐴=πœ‹π‘…2⇒𝐡axisβ‰ˆπœ‡04πœ‹2πœ‡π‘Ÿ3
Similar to electric field Let electric dipole moment 𝑝=π‘žπ‘ β‡’πΈaxisβ‰ˆ14πœ‹πœ€02π‘π‘Ÿ3

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


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