B-field lines always form complete loops

• Thus the net flux is 0
• So Gauss’ Law for magnetic fields is

$$ \oint_S\vec B\cdot\text d\vec A=0 $$

❕There is no magnetic monopoles

Circulation of B-field’s

$$ \oint_C\vec B\cdot\text d\vec l=\mu_0 I_\text{enc}\left ( =\mu_0\int_S\vec j\cdot \text d \vec A \right) $$

• $\oint_C$ is the closed line integral around curve $C$
• $I_\text{enc}$ is the current enclosed with the curve $C$
• $\vec j$ is the current density vector
• $S$ is the surface bounded by the curve $C$

Circulation for infinitely long wire

We can use Biot-Savart formula for infinitely long wire

$$ \vec B(s)=\frac{\mu_0 I\,\hat\theta}{4\pi}\left[ \frac{\sin\alpha}{s}\right]^\frac{\pi}{2}_{-\frac{\pi}{2}}=\frac{\mu_0 I}{2\pi s}\,\hat\theta $$

• We take an Ampèrian loop $C$ as a circle centered on the wire
• We can derive the following properties:

$$ \text d\vec l=\text dl\,\hat \theta\Rightarrow \vec B\cdot\text d\vec l=B\,\text dl \quad ; \quad |\vec B(s)|=\text{constant} $$

• so:

$$ \oint_C\vec B\cdot\text d\vec l=\oint_CB\,\text dl=B\oint_C\text dl=B2\pi s=\mu_0I $$

Solenoids

<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/94e94907-9fb8-4f3b-ad06-3c81a53fa029/Solenoid.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/94e94907-9fb8-4f3b-ad06-3c81a53fa029/Solenoid.png" width="40px" /> A Solenoid is a helical coil of wire

• $N$ turns ; length $L$ ; $n=N/L$ turns per unit length

</aside>

• For a long solenoid, B-fields from each loop superpose and sum inside
  • $B_\text{in}\approx \text{uniform}$
  • $B_\text{out}\approx 0$

B-field in a solenoid

$$ \begin{aligned} \oint_C\vec B\cdot \text d\vec l&=B_\text{in}l=\mu_0I_\text{enc}=\mu_0nlI \\ &\Rightarrow B_\text{in}=\mu_0nI

\end{aligned} $$