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Derivation of Biot Savart law in College Physics

2022-07-06 01:57:00 【Yingzhi 123】

Biot · Biot Savart's Law

$dB=\frac{\mu_0}{4\pi} \frac{Idle_r} {r^2}$

among e_r Is the unit vector from the current element to the direction of the reference point , $\mu _0$ Is the vacuum permeability . The magnetic induction intensity of the magnetic field generated by the current element vertical Idl And e_r A plane composed of , And meet the right-hand screw rule .

Current element

Definition ： $I\vec{d}l$ Is the current element . The size is Idl , $\vec{d}l$ The direction of is determined by the current flow direction where the line element is located .

Purpose ： Use the integral method to calculate the magnetic field distribution of any shape .

Magnetic field of current element

size ： $dB=\frac{\mu_0}{4\pi}\frac{I\vec{d}l\times \vec{r}}{r^3}=\frac{\mu_0}{4\pi} \frac{Idlsin\theta } {r^2}$

Magnetic field of current carrying straight wire

Long for Current carrying straight conductor , The current is , The calculated distance from the straight wire is r_0 Of The magnetic induction intensity of the point .

The mathematical formula involved

$cot\theta =\frac{cos\theta}{sin\theta}$

${(cot\theta)}'=\frac{{(cos\theta)}'}{{sin\theta}}-\frac{cos\theta{(sin\theta)}'}{(sin^2\theta)}=-\frac{1}{sin^2\theta}$

$l=r_0cot(\pi-\theta)=-r_0cot\theta$

Integral derivation of magnetic induction intensity

$B= \int dB=\frac{\mu_0}{4\pi}\int_{C}^{D} \frac{Idlsin\theta } {r^2}$

$\frac{dl}{d\theta}=l'=-r_0(cot\theta)'=\frac{r_0}{sin^2\theta}$

$r=\frac{r_0}{sin(\pi-\theta)}=\frac{r_0}{sin\theta}$

therefore ：

$B=\frac{\mu_0I}{4\pi}\int_{\theta_1}^{\theta_2} \frac{sin\theta } {r_0^2}d\theta=\frac{\mu_0I}{4\pi r_0}(cos\theta_1-cos\theta_2)$

Infinite current carrying straight wire

be $\theta_1 = 0$ , $\theta_2=\pi$

$B=\frac{\mu_0I}{2\pi r_0}$

Expanding knowledge

Magnetic phenomenon

All magnetic phenomena originate from the motion of electric charges .
All magnetic forces are essentially the force between charges .

There are four basic forces in the universe

1、 Gravity is also called gravity , It is the weakest of the four basic interactions , But at the same time, it has the largest scope of action . In general relativity, gravity is due to curved space and time .

2、 Electromagnetic force ： Most substances in the world have electromagnetic force , And magnetism and electricity are one of the expression modes of electromagnetic force .

3、 Strong interaction force is also called strong nuclear force , All matter in the universe is made up of atoms , Atoms are made up of electrons and nuclei , The nucleus is made up of neutrons and protons .

4、 Weak interaction force is also called weak nuclear force , It can be said to be another source of nuclear energy , It is mainly the natural radiation produced by the nucleus , Among the four interaction forces , Weak interaction is only a little stronger than gravity .

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