2024-02-26 Magnetic Field

Review

Introductory physics introduced the following:

• Sources of magnetic field (current)
• Magnetic/Lorentz force $\mathbf{F}=q\mathbf{v}\times \mathbf{B}$
• Ampere’s law in integral form
• Biot-Savart law for line currents

Useful Relations: Straight line segment magnetic field

$$B=\frac{{\mu }_{0}I}{2\pi r}$$

At the center of a circular loop of radius R:

$$B=\frac{{\mu }_{0}I}{2R}$$

New material:

• Magnetostatic Maxwell Equations in differential form
• Biot Savart law in general form
• Magnetic vector potentials

Differential form of Magnetostatic equations

$$\nabla \cdot \mathbf{B}=0$$ $$\nabla \times \mathbf{B}={\mu }_0\mathbf{J}$$

• Looking at Maxwell’s equations in integral form we can derive the above equations using Divergence Theorem and Stokes Theorem, similarly to electrostatics.

The derivations are below:

$$\oint \vec{B}\cdot d\vec{A}=\int (\vec{\nabla }\cdot \vec{B})dV\to \vec{\nabla }\cdot \vec{B}=0$$ $$\oint \vec{B}\cdot d\vec{s}=\int (\vec{\nabla }\times \vec{B})\cdot d\vec{A}={\mu }_0{i}_{\text{enc}}={\mu }_0\int \vec{j}\cdot d\vec{A}$$ $$\int (\vec{\nabla }\times \vec{B}-{\mu }_0\vec{j})d\vec{A}=0\to \vec{\nabla }\times \vec{B}={\mu }_0\vec{j}$$

Continuity Equation

Describes the behavior of current flowing through a volume.

$$\vec{\nabla }\cdot \vec{j}=-\frac{\partial s}{\partial t}$$

General Biot-Savart Law

$$\vec{\mathbf{B}}(\vec{r})=\frac{{\mu }_0}{4\pi }\int \frac{\vec{j}({\vec{r}}^{\prime })\times |\vec{r}-{\vec{r}}^{\prime }}{|\vec{r}-{\vec{r}}^{\prime }{|}^2}d{V}^{\prime }$$

Electric Current Densities

Total current is integral of current density integrated over surface

$$\vec{j}⟹I=\int \vec{j}\cdot d\vec{A}$$

Magnetic Vector Potential

The magnetic field can always be obtained as the curl of a vector function that we call the magnetic vector potential

$$\vec{\mathbf{B}}=\vec{\nabla }\times \vec{A}$$

Where $\vec{A}$ is the magnetic vector potential

$$\vec{\nabla }\cdot \vec{B}=0⟹\vec{\nabla }\cdot (\vec{\nabla }\times \vec{A})=0$$

The divergence of the curl of a vector is 0 as per the book.

$$\vec{\nabla }\times \vec{B}={\mu }_0\vec{j}⟹\vec{\nabla }\times \vec{\nabla }\times \vec{A}=\vec{\nabla }(\vec{\nabla }\cdot \vec{A})-{\vec{\nabla }}^2\vec{A}={\mu }_0\vec{j}$$ $$\vec{\nabla }\cdot \vec{A}=0$$

The above equation is also known as the coulomb gauge

General Field Theory

$$\vec{\nabla }=-\vec{\nabla }\varphi +\vec{\nabla }\times \vec{A}$$ $$\vec{\nabla }\cdot \vec{V}=-{\vec{\nabla }}^2\varphi +\vec{\nabla }\cdot (\vec{\nabla }\times \vec{A})$$

Plugging in potential

$$=-{\vec{\nabla }}_{\vec{r}}^2\varphi =\frac{1}{4\pi }{\int }_{\vec{V}}S({\vec{r}}^{\prime }){\vec{\nabla }}_{\vec{r}}^2\frac{1}{|\vec{r}-{\vec{r}}^{\prime }|}d{V}^{\prime }$$ $$=-\frac{1}{4\pi }\int S({\vec{r}}^{\prime })(-4\pi \delta (\vec{r}-{\vec{r}}^{\prime }))d{V}^{\prime }$$

Curl

$$\vec{\nabla }=-\vec{\nabla }\varphi +\vec{\nabla }\times \vec{A}$$ $$\vec{\nabla }\times \vec{\nabla }=\vec{\nabla }\times (-\vec{\nabla }\varphi )+\vec{\nabla }\times \vec{\nabla }\times \vec{A}$$ $$=\vec{\nabla }(\vec{\nabla }\cdot \vec{A})-{\vec{\nabla }}_{\vec{r}}^2\left(\frac{1}{4\pi }\int \frac{\vec{c}({\vec{r}}^{\prime })}{|\vec{r}-{\vec{r}}^{\prime }|}d{V}^{\prime }\right)$$ $$\vec{\nabla }\times \vec{V}=\vec{\nabla }(\vec{\nabla }\cdot \vec{A})+\vec{C}(\vec{r})=\vec{C}(\vec{r})⟹\vec{\nabla }\cdot \vec{A}=0$$

Solution to DiffEq

$$-\vec{\nabla }\vec{A}={\mu }_0\vec{j}$$ $$\vec{A}=\frac{{\mu }_0}{4\pi }\int \frac{\vec{j}({\vec{r}}^{\prime })}{|{\vec{r}}_0{\vec{r}}^{\prime }|}d{V}^{\prime }$$ $$\vec{B}=\vec{\nabla }\times \vec{A}$$

In class problems

a.

$$K=\frac{I}{2\pi a}$$

b.

$$J(s)=\frac{k}{s}=\frac{I}{2\pi as}$$

a.

$$\oint B\cdot dl={\mu }_{0}I$$ $$Br={\mu }_{0}I\to B_{\text{r>a}}=\frac{{\mu }_{0}I}{r}$$ $$B_{a>r}={\mu }_{0}Ir$$