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

Closed Line Integrals: Work Around a Loop

Why does a field tangent to a circle with constant strength make a closed line integral just field times circumference?

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Your summary note

1. The closed line integral $\oint_{C} \mathbf{F} \cdot d\mathbf{s}$
  Define a closed path as one that ends where it begins and bounds a surface, write the work around it with the circled integral sign in words, and record that a conservative force gives $W = 0$ .
2. The whirlpool force $\mathbf{F} = +k\,r^{-1}\,\hat{\mathbf{u}}_\theta$ around a circle
  Integrate $\mathbf{F} \cdot d\mathbf{s}$ around the circle of radius $r_0$ from $\theta = 0$ to $2\pi$ to reach $W = +2\pi k$ , and record how the counterclockwise convention fixes both the limits and the sign.
3. Tangential field of constant magnitude: $F_\theta \cdot 2\pi r_0$
  Write the general circular result for an azimuthal, axially symmetric field with $F_r = 0$ , and state it as field magnitude times circumference.
4. Magnetic field of a straight wire: $\oint \mathbf{B} \cdot d\mathbf{s} = 2\pi r B_\theta$
  Sketch the circular $\mathbf{B}$ lines around a current-carrying wire, write the closed line integral around a circle of radius $r$ , note its link to Ampère's law, and record that the result is not work.
Attempt 1 of 2