## Small Group Activity: The Wire

Vector Calculus II 2021
Students compute a vector line integral, then investigate whether this integral is path independent.
What students learn
• Practice evaluating line integrals;
• Practice choosing appropriate coordinates and basis vectors;
• Introduction to the geometry behind conservative vector fields.

Consider the vector field given by ($\mu_0$ and $I$ are constants): $\boldsymbol{\vec{B}} = {\mu_0 I\over2\pi} \left({-y\,\boldsymbol{\hat{x}}+x\,\boldsymbol{\hat{y}}\over x^2+y^2}\right) = {\mu_0 I\over2\pi} \, {\boldsymbol{\hat{\phi}}\over r}$
$\boldsymbol{\vec{B}}$ is the magnetic field around a wire along the $z$-axis carrying a constant current $I$ in the $z$-direction.

• Determine $\boldsymbol{\vec{B}}\cdot d\boldsymbol{\vec{r}}$ on any radial line of the form $y=mx$, where $m$ is a constant.
• Determine $\boldsymbol{\vec{B}}\cdot d\boldsymbol{\vec{r}}$ on any circle of the form $x^2+y^2=a^2$, where $a$ is a constant.
You may wish to express the equations for these curves in polar coordinates.

Go: For each of the following curves $C_i$, evaluate the line integral $\int\limits_{C_i}\boldsymbol{\vec{B}}\cdot d\boldsymbol{\vec{r}}$.

• $C_1$, the top half of the circle $r=5$, traversed in a counterclockwise direction.
• $C_2$, the top half of the circle $r=2$, traversed in a counterclockwise direction.
• $C_3$, the top half of the circle $r=2$, traversed in a clockwise direction.
• $C_4$, the bottom half of the circle $r=2$, traversed in a clockwise direction.
• $C_5$, the radial line from $(2,0)$ to $(5,0)$.
• $C_6$, the radial line from $(-5,0)$ to $(-2,0)$.

FOOD FOR THOUGHT

• Construct closed curves $C_7$ and $C_8$ such that this integral $\int\limits_{C_i}\boldsymbol{\vec{B}}\cdot d\boldsymbol{\vec{r}}$ is nonzero over $C_7$ and zero over $C_8$.
It is enough to draw your curves; you do not need to parameterize them.
• Ampère's Law says that, for any closed curve $C$, this integral is ($\mu_0$ times) the current flowing through $C$ (in the $z$ direction). Can you use this fact to explain your results to part (a)?
• Is $\boldsymbol{\vec{B}}$ conservative?

#### Main ideas

• Calculating (vector) line integrals.
• Use what you know!

#### Prerequisites

• Familiarity with $d\boldsymbol{\vec{r}}$.
• Familiarity with “Use what you know” strategy.

#### Warmup

This activity should be preceded by a short lecture on (vector) line integrals, which emphasizes that $\int_C\boldsymbol{\vec{F}}\cdot d\boldsymbol{\vec{r}}$ represents chopping up the curve into small pieces. Integrals are sums; in this case, one is adding up the component of $\boldsymbol{\vec{B}}$ parallel to the curve times the length of each piece.

#### Props

• whiteboards and pens

#### Wrapup

Emphasize that students must express everything in terms of a single variable prior to integration.

Point out that in polar coordinates (and basis vectors) \begin{eqnarray*} \boldsymbol{\vec{B}}= {\mu_0 I\over2\pi} {\boldsymbol{\hat{\phi}}\over r} \end{eqnarray*} so that using $d\boldsymbol{\vec{r}} = dr\,\boldsymbol{\hat{r}} + r\,d\phi\,\boldsymbol{\hat{\phi}}$ quickly yields $\boldsymbol{\vec{B}}\cdot dd\boldsymbol{\vec{r}}$ along a circular arc (${\mu_0 I\over2\pi}\,d\phi$) or a radial line ($0$), respectively.

### Details

#### In the Classroom

• Sketching the vector field takes some students a long time. If time is short, have them do this before class, or consider using MATLAB or similar technology to plot the field. Still, it's important to plot a few vectors by hand.
• Students who have not had physics don't know which way the current goes; they may need to be told about the right-hand rule.
• Some students may confuse the wire with the paths of integration.
• Students working in rectangular coordinates often get lost in the algebra of Question 2b. Make sure that nobody gets stuck here.
• Students who calculate $\boldsymbol{\vec{B}}\cdot d\boldsymbol{\vec{r}}={dy\over x}$ on a circle need to be reminded that at the end of the day a line integral must be expressed in terms of a single variable.
• Some students will be surprised when they calculate $\boldsymbol{\vec{B}}\cdot d\boldsymbol{\vec{r}}=0$ for radial lines. They should be encouraged to think about the directions of $\boldsymbol{\vec{B}}$ and $d\boldsymbol{\vec{r}}$.
• Most students will either write everything in terms of $x$ or $y$ or switch to polar coordinates. We discuss each of these in turn.
• This problem cries out for polar coordinates. Along a circular arc, $r=a$ yields $x=a\cos\phi$, $y=a\sin\phi$, so that $d\boldsymbol{\vec{r}}=-a\sin\phi\,d\phi\,\boldsymbol{\hat{x}}+a\cos\phi\,d\phi\,\boldsymbol{\hat{y}}$, from which one gets $\boldsymbol{\vec{B}}\cdot d\boldsymbol{\vec{r}} = {\mu_0 I\over2\pi}\,d\phi$.
• Students who fail to switch to polar coordinates can take the differential of both sides of the equation $x^2+y^2=a^2$, yielding $x\,dx+y\,dy=0$, which can be solved for $dx$ (or $dy$) and inserted into the fundamental formula $d\boldsymbol{\vec{r}}=dx\,\boldsymbol{\hat{x}}+dy\,\boldsymbol{\hat{y}}$. Taking the dot product then yields, $\boldsymbol{\vec{B}}\cdot d\boldsymbol{\vec{r}}= {\mu_0 I\over2\pi} {dy\over x}$. Students may get stuck here, not realizing that they need to write $x$ in terms of $y$. The resulting integral cries out for a trig substitution --- which is really just switching to polar coordinates.
In either case, sketching $\boldsymbol{\vec{B}}$ should convince students that $\boldsymbol{\vec{B}}$ is tangent to the circular arcs, hence orthogonal to radial lines. Thus, along such lines, $\boldsymbol{\vec{B}}\cdot d\boldsymbol{\vec{r}}=0$; no calculation is necessary. (This calculation is straightforward even in rectangular coordinates.)
• Watch out for folks who go from $r^2=x^2+y^2$ to $d\boldsymbol{\vec{r}} = 2x\,dx\,\boldsymbol{\hat{x}} + 2y\,dy\,\boldsymbol{\hat{y}}$.
• Working in rectangular coordinates leads to an integral of the form $\int-{dx\over y}$, with $y=\sqrt{r^2-x^2}$. Maple integrates this to $-\tan^{-1}\left({x\over y}\right)$, which many students will not recognize as the polar angle $\phi$. If $r=1$, Maple instead integrates this to $-\sin^{-1}x$; same problem. One calculator (the TI-89?) appears to use arcsin in both cases.

#### Subsidiary ideas

• Independence of path.

#### Homework

• Any vector line integral for which the path is given geometrically, that is, without an explicit parameterization.

#### Essay questions

• Discuss when $\oint\boldsymbol{\vec{B}}\cdot d\boldsymbol{\vec{r}}$ around a closed curve will or will not be zero.

#### Enrichment

• This activity leads naturally into a discussion of path independence.
• Point out that $\boldsymbol{\vec{B}}\sim\nabla\phi$ everywhere (except the origin), but that $\boldsymbol{\vec{B}}$ is only conservative on domains where $\phi$ is single-valued.
• Discuss winding number, perhaps pointing out that $\boldsymbol{\vec{B}}\cdot d\boldsymbol{\hat{r}}$ is proportional to $d\phi$ along any curve.
• Discuss Ampère's Law, which says that $\oint\!\boldsymbol{\vec{B}}\cdot d\boldsymbol{\vec{r}}$ is ($\mu_0$ times) the current flowing through $C$ (in the $z$ direction).

Keywords
Line integrals conservative vector fields Ampere's Law simply-connectedness
Learning Outcomes