Line and surface integrals
The work done by a force along a path and the flux of a field through a surface are the two integrals in which electromagnetism is written.
01Integrating along a curve
Until now we integrated over intervals and over regions. Now the domain is a curve. If what is being integrated is a scalar field —a density, for example—, the line integral gives the total quantity along the wire:
With f = 1 you get the length of the curve; with f = linear density, the mass of a wire; with f = charge density, the total charge of a thin-wire conductor.
02Circulation: the work done by a field
If what is being integrated is a vector field, only the component along the path counts. A dot product appears, and the result is the work —or the circulation—:
Reversing the direction of travel changes the sign. And when the curve is closed, it is written with a circle on the integral sign: that is the circulation, which in electromagnetism appears in Ampère’s law and in Faraday’s law.
The three paths join the same endpoints. With a conservative field they give the same result; with a nonconservative one, they do not.
03Path independence
If F = ∇f, then for any path C going from A to B:
It is the vector-calculus counterpart of the Fundamental Theorem of Calculus: only the endpoints matter. And if the path is closed, A = B and the integral equals zero.
That is why, in a DC circuit, Kirchhoff’s voltage law says that the sum of voltages around a closed loop is zero: the electrostatic field is conservative. In contrast, when there is a time-varying magnetic flux, the circulation of E around a loop is not zero, and its value is the induced EMF: that is precisely Faraday’s law.
04Surface integrals and flux
The next step is to integrate over a surface. With a scalar field you get, for example, the mass of a thin sheet; with a vector field, the flux: how much passes through the surface.
| Flux of | Called | Where it appears |
|---|---|---|
| Electric field E | Electric flux | Gauss’s law: \( \Phi = \dfrac{Q_{enc}}{\varepsilon_0} \) |
| Magnetic field B | Magnetic flux \( \Phi \), in webers | Faraday’s law: \( \text{EMF} = -\dfrac{d\Phi}{dt} \). It is the principle of the transformer |
| Current density J | Current | \( I = \iint \vec{J} \cdot \hat{n} \, dS \) through the cross section of a conductor |
| Poynting vector \( \vec{S} = \vec{E} \times \vec{H} \) | Power | The power carried by a wave through a surface |
A closed surface has its normal pointing outward by convention; an open one requires declaring which is its positive face, and that choice fixes the positive direction of travel around the edge by the right-hand rule. The minus sign in Faraday’s law —Lenz’s law— depends on that convention.
05Where this is used in electronics
- Induced EMF. A transformer works because a time-varying magnetic flux through the secondary induces a circulation of electric field: the output voltage. The whole calculation is a surface integral differentiated in time.
- Inductance. It is defined as flux linkage over current, L = Φ/I: another surface integral.
- Current through a conductor. If the current density is not uniform —skin effect at high frequency— the total current is the integral of J over the cross section.
- Power radiated by an antenna. It is the flux of the Poynting vector through a sphere surrounding it. From there comes the radiation pattern of antennas.
- Thermal dissipation. The heat leaving a heat sink is the flux of the heat-flux vector through its surface.
06In the lab
Compute by hand ∫F·dr for F = (y, x) from (0,0) to (2,1), first along the straight segment and then along the L-shaped path. Verify that they agree and find the potential f = xy that explains it.
With a coil and a magnet, measure the induced voltage when moving it at different speeds. Relate the reading to −dΦ/dt and verify that the sign changes when the motion is reversed.
Compute the flux of a uniform field through a flat loop for three different angles, and check experimentally that the induced EMF when rotating it is maximum when the plane is parallel to the field.
07Common mistakes
- Forgetting the factor |r′(t)| in the line integral of a scalar field.
- Confusing ∫f ds with ∫F·dr: the first does not depend on the direction, the second does.
- Assuming that work does not depend on the path without checking that the field is conservative.
- Forgetting the orientation of the surface and getting the sign of the flux wrong.
- Integrating the magnitude of the field instead of its normal component when computing flux.
08Self-assessment
What does ∫C ds represent with f = 1?
The length of the curve C.
Compute ∫F·dr for F = (y, x) from (0,0) to (2,1) along the straight segment.
Since F = ∇(xy), the integral equals f(2,1) − f(0,0) = 2 − 0 = 2, regardless of the path.
What is the circulation of a conservative field around a closed path?
Zero.
Why is Kirchhoff’s voltage law a case of this?
Because the sum of voltage drops around a loop is the circulation of the electrostatic field, which is conservative and therefore gives zero.
A 5 cm² loop in a perpendicular field of 0.2 T. What is the flux?
Φ = B·A = 0.2 · 5·10−4 = 10−4 Wb. If the field dropped to zero in 1 ms, the average EMF would be 0.1 V.
What does the flux of the Poynting vector measure?
The electromagnetic power crossing the surface under consideration.
09Further reading
- James Stewart. Multivariable Calculus. 7th ed. (Spanish edition, “Cálculo de varias variables. Trascendentes tempranas”), Cengage Learning, 2012. Chapter 16 goes through line integrals, path independence and surfaces.
- Jerrold E. Marsden and Anthony J. Tromba. Vector Calculus. 5th ed. (Spanish edition, “Cálculo vectorial”), W. H. Freeman, 2003. With formal care about the orientation and parametrization of surfaces.
- Matthew N. O. Sadiku. Elements of Electromagnetics. Spanish edition, “Elementos de electromagnetismo,” 5th ed., Alfaomega, 2018. The same integrals, but always posed from a field problem.