Electricity and Magnetism

Mathematical Tools and Fields

01Series Approximations: Keeping One or Two Terms 02Integrals as Sums: Along a Line and a Curved Path 03Line Integrals: Work as the Integral of F · ds 04Closed Line Integrals: Work Around a Loop 05Double Integrals: Area, Charge and Volume 06Closed Surface Integrals: The Sphere and Its Charge 07Electric Flux Through Surfaces and Volume Integrals Problem set0/10 Problem set 20/10 MIT problem set0/4

01Field Theory Versus Action at a Distance 02Scalar Fields: Contour, Color and Relief Maps 03Vector Fields, Sources, Sinks and Divergence 04Curl: Measuring the Circulation of a Field 05Gravitational, Electric and Magnetic Fields 06Drawing Vector Fields: Arrows, Lines and Grass Seeds Problem set0/10 Problem set 20/10 MIT problem set0/1

Line Integrals: Work as the Integral of F · ds

How does the dot product F · ds pick out just the part of a force that does work along a path?

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Your summary note

1. The dot product $\mathbf{F} \cdot d\mathbf{s} = |\mathbf{F}|\,|d\mathbf{s}|\cos\theta$
  Write the definition as the work element $dW = F_{\parallel}\,|d\mathbf{s}|$ , sketch the force, displacement and angle $\theta$ , and record the Cartesian form $F_x\,dx + F_y\,dy + F_z\,dz$ .
2. The polar form $F_r\,dr + F_\theta\,r\,d\theta$
  Sketch the radial and angular components of both $\mathbf{F}$ and $d\mathbf{s}$ , derive the arc length $r\,d\theta$ for the tangential step, and state that a central force has $F_\theta = 0$ .
3. Reducing $U(b) - U(a) = -\int_a^b \mathbf{F} \cdot d\mathbf{s}$ to one ordinary integral
  Write the general definition, then reduce it to $-\int_a^b F_x\,dx$ when $dy = dz = 0$ and to $-\int_a^b F_r\,dr$ when $F_\theta = 0$ , noting which point sits in each limit.
4. Worked example: repulsive inverse-square force, $U(r) = k/r$
  Integrate $-\int_\infty^r k\,r^{-2}\,dr$ with $U(\infty) = 0$ , check that the positive sign matches pushing in against repulsion, and write the analogue $V(b) - V(a) = -\int_a^b E_r\,dr$ in volts.
Attempt 1 of 2