• Use Stokes’ theorem to relate line integrals to surface integrals
• Use Stokes’ theorem to calculate a surface integral
• Use Stokes’ theorem to calculate a curl

Analyzing Electromagnetic Induction in a Wire Loop

You are learning electronics repair at a local makerspace and currently studying electromagnetic induction—the principle that makes transformers, generators, and wireless chargers work. Your mentor has set up a demonstration with a wire loop placed in a changing magnetic field, and you need to analyze the relationship between the electric field induced in the wire and the magnetic flux through the loop. Understanding how Stokes’ theorem connects the circulation of the electric field around the wire to the curl of the field over the surface helps you predict voltage outputs and troubleshoot faulty coils. Today, you will work through several calculations involving vector fields, verifying relationships between line integrals and surface integrals, and applying Faraday’s law to determine how electric fields respond to changing magnetic fields.

Consider a vector field representing a simplified electric field model: [latex]{\bf{F}}(x,y,z)=\langle 2z, x, y \rangle[/latex]. You need to verify Stokes’ theorem by computing both the line integral around a boundary curve and the surface integral of the curl over a surface.

Your mentor presents a more complex surface: a wire loop forms the boundary of a paraboloid cap. Consider vector field [latex]{\bf{F}}(x,y,z) = \langle y, -x, z^2 \rangle[/latex] and surface [latex]S[/latex], which is the part of the paraboloid [latex]z = 9 - x^2 - y^2[/latex] that lies above the [latex]xy[/latex]-plane, oriented upward. The boundary curve [latex]C[/latex] is the circle [latex]x^2 + y^2 = 9[/latex] in the [latex]xy[/latex]-plane.

Rather than computing a complicated surface integral, use Stokes’ theorem to evaluate [latex]\displaystyle\iint_S \text{curl }{\bf{F}} \cdot d{\bf{S}}[/latex] by instead computing the line integral [latex]\displaystyle\int_C {\bf{F}} \cdot d{\bf{r}}[/latex].

An important consequence of Stokes’ theorem is surface independence: if two surfaces share the same boundary curve and orientation, then the flux integral of the curl is the same over both surfaces.

Consider vector field [latex]{\bf{G}}(x,y,z) = \langle z, x, y \rangle[/latex]. Surface [latex]S_1[/latex] is the hemisphere [latex]z = \sqrt{1 - x^2 - y^2}[/latex], and surface [latex]S_2[/latex] is the flat disk [latex]x^2 + y^2 \leq 1[/latex] in the plane [latex]z = 0[/latex]. Both surfaces are oriented upward and share the same boundary: the unit circle in the [latex]xy[/latex]-plane.

Now your mentor explains Faraday’s law of electromagnetic induction. In the makerspace, you’re examining a coil where the magnetic field is changing over time.

The magnetic field is given by [latex]{\bf{B}}(t) = \langle 0, 0, 5t^2 \rangle[/latex], where [latex]t[/latex] represents time in seconds. According to Faraday’s law in differential form, [latex]\text{curl }{\bf{E}} = -\frac{\partial {\bf{B}}}{\partial t}[/latex].

Your mentor shows you a scenario where the magnetic field is constant: [latex]{\bf{B}} = \langle 2, -3, 7 \rangle[/latex]. This represents a static magnetic field that does not change over time.