Stokes' Theorem

Prerequisites

• Definition of Curl (for deriving Stokes' Theorem)
• Flux (for deriving Stokes' Theorem)
• Integral Version of Ampere's Law (for deriving the differential version)

In-class Content

May be some duplicate lectures here, plus it feels like we need an activity here.

• Reading: GVC § Stokes' Theorem
• Derivation of Stokes' Theorem (lecture). We follow “div, grad, curl and all that”, by Schey
• Stokes' Theorem (lecture) (Math 3.12: Stokes' Theorem)
• Differential Form of Ampère's Law: Maxwell Eq. 2 & 4 $\Vec{\nabla } \times \Vec{E} = 0$, $\Vec{\nabla } \times \Vec{B} = \mu_0 \Vec{J}$(lecture) (Physics 41: Differential Form of Ampère's Law)

Homework for Static Fields

1. (StokesVerify) Verify Stoke's Theorem for a given field and a hemispherical surface.

Verify Stokes' Theorem for $\FF( r, \theta, \phi)=e^{r^2} \hat{r} + {1\over 2}\sin\theta \,\hat{\phi}$ where the butterfly net surface is the hemisphere of radius 5 centered at the origin with $z\ge 0$.

Homework for Static Fields

1. (amperelawdifferential)

Find the volume current density that produces the following magnetic field (expressed in cylindrical coordinates):

$\vec{B}(\vec{r})=\begin{cases} \frac{\mu_0\,I\,s}{2\pi a^2}\hat{\phi}& s\leq a \\ \frac{\mu_0\,I}{2\pi s}\hat{\phi}& a<s<b \\ 0& s>b \\ \end{cases}$

What is a physical situation that corresponds to this current density?

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