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# Lecture 11: Stokes Theorem - PowerPoint PPT Presentation

Lecture 11: Stokes Theorem. Consider a surface S , embedded in a vector field Assume it is bounded by a rim (not necessarily planar) For each small loop For whole loop (given that all interior boundaries cancel in the normal way).

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Presentation Transcript

• Consider a surface S, embedded in a vector field

• Assume it is bounded by a rim (not necessarily planar)

• For each small loop

• For whole loop (given that all interior boundaries cancel in the normal way)

OUTER RIM SURFACE INTEGRAL OVER ANY

SURFACE WHICH SPANS RIM

• Curl is incompressible

• Consider two surfaces (S1 + S2) with same rim

must be the the same for both surfaces (by Stokes Theorem)

• For closed surface formed by

• Also for a conservative field

using the divergence theorem

by Stokes Theorem

CONSERVATIVE = IRROTATIONAL

• These are examples (see Lecture 12) of second-order differential operators in vector calculus

• So this is a conservative field, so we should be able to find a potential 

• All can be made consistent if

z

(0,0,1)

C2

quarter circle

C1

C3

y

(0,1,0)

x

• Check via direct integration

• Magnetic Field due to a steady current I

• “Ampere’s Law”

• Since

• obeys a continuity equation with no sources or sinks (see Lecture 13)

applying Stoke’s Theorem

units: A m-2

Differential form of Ampere’s Law

z axis along the wire, r is radial distance from the z axis

|B|

a r

• Outside wire: z component from loops in the xy plane

• z component is zero, because

• and x and y components are also zero

in the yz and xz planes

• Therefore

R2

d

R1

• Note that region where field is conservative ( ) is not simply connected: loop cannot be shrunk to zero without going inside wire (where field is not conservative).