Electromagnetism Physics 15b Lecture #5 Curl Conductors Purcell 2.13 3.3 What We Did Last Time Defined divergence: Defined the Laplacian: From Gauss s Law: Laplace s equation: F da divf = lim S V 0 V Guass s Law (local version): E = 4πρ Linked to the integral version by the Divergence Theorem: F da = F S dv V 2 f = f 4πρ = 2 φ 2 φ = 0 Average theorem, no-max/min theorem, impossibility theorem, uniqueness theorem ( ) = 2 f = F = F x x + F y y + F z z x + 2 f 2 y + 2 f 2 z 2 E = ρ ˆr dv 2 r 4πρ = E Electric Field E Charge ρ φ Electric density φ = ρ potential r dv 4πρ = 2 φ E = φ φ = E ds 1
Today s Goals Define curl of vector field Start from line integral around a loop and shrink Stokes Theorem A few examples to familiarize ourselves Discuss conductors Finally a real stuff??? Consider good conductors holding electric charges in a static condition How do the charges distribute? What electric field, potential should appear? What do you mean by good? George Gabriel Stokes (1819 1903) 3-D, 2-D, 1-D, 0-D We have V P 2 FdV = F da V S f ds = f (P 2 ) f (P 1 ) 3-D integral linked to 2-D integral over the boundary 1-D integral linked to values at end points (= 0-D) Shouldn t there be a similar relation between 2-D and 1-D? It could be a surface integral and a line integral around it s border P 2 P 1 P 1 2
Circulation Consider a closed (loop) path C in space where a vector field F exists C Define circulation as the line integral Γ = C F ds This is zero for static electric field ds Split C into two sub-loops C 1 and C 2 with a bridge The integrals along the bridge cancel Γ = Γ 1 + Γ 2 = F ds + F ds C2 C 1 We can continue splitting into ever smaller loops: Γ = j Γ j C 1 C 2 F Curl As we make the area a j of each loop smaller, the line integral Γ j shrinks as well F ds Cj C j The ratio Γ j /a j remains finite lim a j 0 a j a j For a very small loop, the shape does not matter, but the orientation does Define the orientation vector ˆn of the loop ŷ by the right-hand rule At a given point, draw three loops with ˆn = ˆx, ŷ, ẑ Calculate the limits and build a vector ẑ curl of F Curl F is a vector field, representing the circulation per unit area in three possible orientations ˆx 3
Stokes Theorem Going back to splitting circulation, in the limit of small loops F ds = lim F ds C = lim curlf a Cj j = curlf da S a j 0 This is Stokes Theorem Like Gauss s Divergence Theorem, this is math, not physics Applying our (physics) knowledge that line-integral of electric field is path independent, we get E ds C So far so good j a j 0 = 0 for any closed path C curle = 0 Q: What exactly is a curl and how do I calculate it? j Curl in Water You are swimming in a river, with velocity of water flow v If you swim in a loop and come back to the starting point, are you helped, or impeded, by the flow? That s circulation v ds > 0 helped, < 0 impeded C If you drop a small leaf in the v water, will it rotate? That s curl curlf 0 rorates If you drop a thousand leaves everywhere, you can tell which way you should swim by Stokes Theorem v 4
Cartesian Curl Focus on the z-component of curl F Draw a rectangular loop on the x-y plane For small Δx, Δy, we can approximate the line-integral on each side by (length) x (value of F in the middle) F ds = ΔxF C x (x + Δx,y,z) + ΔyF Δy (x + Δx,y +,z) 2 y 2 ΔxF x (x + Δx,y + Δy,z) ΔyF Δy (x,y +,z) 2 y 2 = Δx F x (x + Δx,y,z) F Δx (x + 2 x 2 = ΔxΔy F x y + F y x Dividing by the area ΔxΔy,y + Δy,z) + Δy F y x- and y-components are similar ( curlf) ẑ = F y z y ˆn = ẑ Δx (x,y,z) Δy Δy (x + Δx,y +,z) F Δy (x,y + 2 y x F x y x,z) 2 Cartesian Curl curlf = ˆx F z y F y z + ŷ F x z F z x + ẑ F y x F x y = F Some people find it easier to remember this as a determinant: curlf = ˆx ŷ ẑ x y z Spherical: F x F y F z Cylindrical: curlf = 1 F z r φ F φ z ˆr + F r z F z r ˆφ + 1 r ( rf φ ) F r r φ ( ) r ẑ 1 (F φ sinθ) curlf = F θ r sinθ θ φ ˆr + 1 1 F r r sinθ φ (rf ) φ r ˆθ + 1 rf θ F r r θ ˆφ 5
Examples F y = ax ( F) z = F y x F x y = a F x = ay, F y = ax ( F) z = 2a ay F x = x 2 + y 2 ax F y = x 2 + y 2 F = 0 Curl of Gradient The curl of the gradient of a scalar field is ( ) ( f ) = 0 Check explicitly: ( ( f )) ˆx = f y x f x y = 0 Use Stokes Theorem: ( ( f )) da = S f ds = 0 C If a vector field (like E) can be expressed as a gradient of a scalar field (like ϕ), it must be curl-free ( ) On the other hand, any curl-free vector field can be expressed as a gradient of a scalar field If F = 0, by Stokes it s line-integral must be path-independent Line integral from an arbitrary chosen reference point will do r f (r) = F ds F = f (r) 0 6
3-D, 2-D, 1-D, 0-D surface volume FdV = F da volume surface F = F x x + F y y + F z z surface loop FdV = F ds surface loop F = ˆx F z y F y z +ŷ F x z F z x +ẑ F y x F x y P 1 P 2 f ds = f (P 2 ) f (P 1 ) P 1 curve P 2 f = ˆx f x + ŷ f y + ẑ f z Conductor Conductor vs. insulator is a matter of speed When electric field is applied to any material, charged particles inside move Good conductor: many charges, quick to move Good insulator: few charges, slow to move Conductivity = a measure of goodness of conductor σ = n v E Unit in CGS: (charge density) (average velocity) = (applied electric field) (esu cm 3 ) (cm s) dyne esu Will define this properly in Lecture 8 = 1, in SI: s (C m 3 ) (m s) V m = 1 Ω m 7
Conductivity Conductivity σ of material span 23 orders of magnitude They fall into three broad groups: σ >> 1 conductors Mostly metals, in which free electrons act as the charge carriers σ = O(1) semiconductors σ << 1 insulartors In electrostatics, where time is not an issue, we only have to consider two extreme cases Perfect conductor : σ = Perfect insulator : σ = 0 Material σ [1/Ωm] silver 6.3 x 10 7 copper 5.9 x 10 7 manganin alloy 2.3 x 10 6 silicon 33 salt water 23 drinking water 0.0005 0.05 deionized water 5.5 x 10-6 wood 10-8 10-11 glass 10-10 10-14 sulfur 5 x 10-16 rubber 10-13 10-16 source: J.E. Hoffman Field Inside Conductor As long as we only consider a static condition ( electrostatic equilibrium ) in which no charge is moving, there is no electric field inside conductors If E 0, then the charges would be still moving When a conductor is placed in an external field E ext, charges move inside because of F = qe charge distribution ρ becomes non-zero internal electric field E int is generated and the process continues until E ext + E int = 0 Sounds a little involved? This is not really different from any system of fluids settling into the lowest-energy state + + + + 8
Potential and Charge Density Since E = 0 inside a conductor, Δφ BA = E ds = 0 A for any two points connected by a conductor A The entire volume (and surface) of a contiguous piece of conductor is an equipotential B Also, there is no net charge density inside a conductor because 4πρ = E = 0 All electric charges, if any, must be on the surface B + + + + NB: There are electrons and protons inside the conductor. Their densities are balanced so that there is no net charge after averaging over small volumes Electric Field Near Surface Consider a conductor with surface charge density σ What is the E field immediately outside? We know the surface is an equipotential E field must be perpendicular to it Draw a cylinder half buried into the surface Charge inside the cylinder is σa E = 0 at the bottom (inside) E is parallel to the side of the cylinder Gauss s Law: 4πσ A = E da = EA E = 4πσ ˆn top where ˆn is to the surface area = A E 9
Summary Defined curl of vector field by In Cartesian: curlf = F = ˆx F z y F y + ŷ F x z z F z x + ẑ F y x F x y Stokes Theorem: F ds curlf ˆn lim C a 0 a FdV = F ds surface loop Studied (ideal) conductor in electrostatic equilibrium E = 0 inside, ϕ = const, no net charge density inside Field immediately outside is perpendicular to the surface and E = 4πσ, where σ is the surface charge density 10