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 \((-2,0)\).


  • 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!


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


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.


  • whiteboards and pens


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.


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.


  • 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.


  • 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).

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