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UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 1 Prof. Steven Errede LECTURE NOTES 1 Introduction: • In this course, we will study/investigate the nature of the ELECTROMAGNETIC INTERACTION (at {very} low energies, i.e. E ~ 0 GeV, {1 GeV = 109 electron volts = 1.602×10−10 Joules}). The electromagnetic interaction is ONE of FOUR known FORCES (or INTERACTIONS) of Nature: 1) Electromagnetic Force – binds electrons & nuclei together to form atoms - binds atoms together to form molecules, liquids, solids. . . . gases 2) Strong Force – binds protons & neutrons together to form nuclei 3) Weak Force – responsible for radioactivity (e.g. β decay) (weak force important @ high energies) 4) Gravity – binds matter together to form stars, planets, solar systems, galaxies, etc. • • At the MICROSCOPIC (i.e. QUANTUM) LEVEL (elementary particle physics) the forces of nature are mediated by the exchange of a “force-carrying” particle e.g. between two “charged” particles: mediating force carrier • charge B •charge A Quantum Field Theory Force QED 1) EM QCD QWD QGD Mass of force mediator Range of force mediator ≡ 0.000 ∞ 2) STRONG Force Mediator single PHOTON octet of Force Type attractive & repulsive attractive & 3) WEAK GLUONs W±, Zo repulsive attractive & repulsive ≡ 0.000 Mw ≈ 80.4 GeV/c2 Mz ≈ 91.2 GeV/c2 attractive only ≡ 0.000 4) GRAVITY single GRAVITON Intrinsic spin of force mediator Charge associated w/ force 1 ±e r, g, b ~ 1fm 1 r , g,b ~ 1fm 1 ± gW ∞ 2 MASS, m (unquantized) At high energies, QED & QWD unify to become a single force, known as the ELECTROWEAK FORCE = Planck’s constant divided by 2π = h/2π = 1.0546 x 10-34 Joule – seconds mproton= 0.93 GeV/c2 = 1.67262158×10−27 kg 1 fm = 1 femto-meter = 1 Fermi = 10-15meters ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. 1 UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 1 Prof. Steven Errede Pattern of Masses for Fundamental, Spin-½ Matter Particles - Fermions u c t have electric charge + 2/3 d s b have electric charge −1/3 “doublets” of quarks: fractional electric charge charge!!! Each quark comes in 3 strong (“color”) charges: red, green, blue “doublet” of anti-quarks: 2 u c t q = -2/3 d s b q = +1/3 with 3 anti-color charges: red, green, blue (i.e. anti-red, anti-green, anti-blue) ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 1 Prof. Steven Errede Questions: Why are there 3 generations of quarks & leptons? Internal Quantum #? Why not just one? Are there more? (seemingly not…) What physics is responsible for the observed pattern of quark/lepton masses? Why are there four forces of nature? Why not just one? Are there more forces? Note that ALL 4 fundamental forces of nature have both electric & magnetic fields!!! “Magnetic” field arises from motion of “electric” charge in space – relativity & space-time involved here! FORCE “ELECTRIC” FIELD “MAGNETIC” FIELD EM STRONG WEAK GRAVITY EM – electric chromo–electric weak–electric gravito–electric EM – magnetic chromo–magnetic weak–magnetic gravito–magnetic Nordvedt Effect e.g. affects motion of moon’s orbit around earth (very small) no motion/movement Electric Field – time-averaged field (macroscopic) present for static charges exchanging virtual quanta associated w/given force Magnetic Field – time averaged field (macroscopic) arises/associated w/moving charges – motional effect Magnetic field arises from motion of charge Any/all/each of 4 fundamental forces of nature any/all/each of 4 fundamental forces of nature Magnetic field results from charge + space-time structure of our universe!! At microscopic level, EM force mediated by (virtual) photons − two electrically charged particles “know” about each other by exchanging virtual photons. Virtual photon • charge e •charge e ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. 3 UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 1 Prof. Steven Errede Planck’s constant ⎛ ⎞ Virtual photons carry linear momentum, pγv ⎜ = h ⎟ λ γv ⎠ ⎝ but have zero total energy: DeBroglie wavelength c = speed of light = 3 × 108 m/sec Eγ2v = pγ2v c 2 + mγ2v c 4 = 0 Real Photons (e.g. visible light): Eγ2R = pγ2R c 2 mγ R = 0 If Eγ v = 0, then pγ2v c 2 = − mγ2v c 4 pγ R = h λ γ , Eγ = hfγ > 0 R R R i.e. pγ = ±imγ v c 2 i= -1 complex! v If Eγ v = 0 then: Eγ v = hfγ v = 0 ⇒ fγ v = 0 virtual photons have zero frequency, but have non-zero DeBroglie wavelength, λγ V > 0 ! FORCE: F = ma (Newton’s 2nd Law) d mγ v vγ v Δp dp = = F = Δt dt dt ( ) = mγ t dvγ r dt = mγ t a electric charges emit & absorb virtual photons (lots of them!!!) − each such photon carries with it momentum, Pγ v − depending on sign of momentum (emitted/absorbed), a net force will result, acting on each charged particle + 1 e1+ Opposite charges – attract: e2+ •e1 Like charges – repulsive: Fe+ • Fe+ 2 − • •e2 Fe+ Fe− 1 2 n.b. Your own body can sense virtual photons!!! o Get your comb out, comb your hair several times - charges up comb via static electricity o Bring comb near to e.g. hair on your forearm & feel the pull on forearm hairs from electric charge on comb (works best in winter/dry conditions). 4 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 1 Prof. Steven Errede ELECTROSTATIC FIELDS IN A VACUUM COULOMB’S LAW It has been experimentally observed (Charles Augustin Coulomb, 1785) that the net, time-averaged force (i.e. summed over many, many virtual photons) between two stationary point charges Qa & Qb: 1) Acts along the line joining the two point charges, Qa & Qb (i.e. radial force!) 2) Is linearly proportional to the product of the two point charges, Qa * Qb (n.b. Force is charge-signed!) – Net force is repulsive if Qa is same sign as Qb. – Net force is attractive if Qa is opposite sign as Qb. 3) Is inversely proportional to the square of the separation distance, rab ≡ rb − ra = rab between the two point charges. The net force exerted by point charge Qa ON point charged Qb is given by: F ab = K Qa Qb rab2 rˆab (SI UNITS – Newtons) constant of unit vector proportionality (points from Qa at ra to Qb at rb ) rab ≡ rb − ra = rab rˆab ≡ rab rab = = rab rab unit vector pointing from point A to point B. Fab is a radial force, one which points from (to) point A to (from) point B, depending on sign of the charge product (QaQb) Qa Qb < 0 is attractive force (F < 0) Qa Qb > 0 is repulsive force (F > 0) ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. 5 UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 1 Prof. Steven Errede The NET force exerted by point charge Qb ON point charge Qa: QQ Fba = K b 2 a rˆba rba Fba is radial force, point from (to) point B to (from) point A, depending on sign of charge product (QaQb) Qa Qb < 0 is attractive force (F < 0) Qb Qa > 0 is repulsive force (F > 0) rba ≡ ra − rb = rba Fab = K Qa Qb r 2 ab rˆba ≡ rba rba = = − rˆab rba rba Fba = K rˆab Qb Qa rba2 Now r ab = r ba, since rab ≡ rb − ra = rab rˆba and rba = ra − rb = rba but note that rba = − rab and/or r ba = − r ab since ( rb − ra ) = − ( ra − rb ) = rab = − rba thus, we see that: Fab = − Fba This is Newton’s 1st Law: For every action, there is equal and opposite reaction. SI units for electric charge Q: Coulombs (C) Fundamental unit of electric charge, Qe = 1.602 x 10−19 Coulombs Question: What is the physics that dictates (specifies/determines) the value of e?? i.e. Why is e = 1.602 x 10−19 Coulombs? What is K? K = 6 1 4πε o in SI units ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 1 Prof. Steven Errede Coulombs 2 Newton -m 2 ⎧ Farads ⎫ ⎨= ⎬ ⎩ meter ⎭ (Farad is SI unit of capacitance) Question: If free space is truly empty, how can it have any measurable physical properties associated with it??? ε o = electric permittivity of free space = 8.8542 x 10−12 Answer: Free space is NOT empty!!! It is “filled” with virtual particle-anti-particle pairs!! (e.g. e+-e−, μ+-μ−, q − q , W+W−, etc. pairs) existing for short time(s), as allowed by the Heisenberg Uncertainty Principle – can “violate” energy (momentum) conservation only for time interval Δt ≤ / ΔE (and over a distance of Δx ≤ / Δpx ). ε0 is the macroscopic, time-averaged (over many many such virtual pairs) electric permittivity of (quantum) vacuum - the physical vacuum behaves like a dielectric medium!!! Thus: Fab = 1 Qa Qb 4πε o rab2 rˆab Fab ẑ QA r ab QB A B ra rb ϑ• r ab = rb − ra = rab ŷ x̂ Factor of 4π = “flux factor” for solid angle associated with flux of virtual photons emitted by point charge!!! Virtual photons “emitted” from QA are emitted into 4π steradians @ point A: γv γv γv γv QA Force decreases as 1 r2 γv Just like/analogous to real γv Photons emitted from e.g. 100 watt light bulb - Intensity decreases as 1 2 from light source. r • γv A γv γv ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. 7 UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 1 Prof. Steven Errede Note the similarity between Coulomb’s Law and Newton’s Law of Gravity: ⎛ 1 ⎞ Qa Qb MaMb rˆ FC = ⎜ ⎟ 2 rˆ ↔ FG = GN r2 ⎝ 4πε o ⎠ r Newton’s constant, GN = 6.673 x 10−11m3kg -1s -2 Or can define: Define: 1 1 1 GE ≡ GN ≡ ε og ≡ g 4πε 0 4 ε0 4π GN FC = GE Qa Qb r 2 rˆ ⎛ 1 ⎞ MaMb then FG = ⎜ rˆ g ⎟ 2 ⎝ 4πε o ⎠ r Coulomb’s Constant Coulomb’s Law FC = 1 Qa Qb 4πε o r2 rˆ “nothing” Note that if dielectric properties of free space (vacuum) were different than they are, then Coulomb’s Law, i.e. the force between electrically charged particles would be different. Consider a universe in which we could change the EM properties of the vacuum at will: im ( ε o → 0 ) : FC → ∞ !! “strong” electromagnetism im ( ε o → ∞ ) : FC → 0 !! “weak” electromagnetism (assuming this doesn’t also affect value of fundamental electric charge, e) Note further/we shall see that: c = speed of light = 1 ε o μo = 3 x 108 m / sec μo = magnetic permeability of free space = 4π x 10−7 Newtons/Ampere 1 Ampere of electric current = 1 Coulomb/sec (I = dQ/dt) Thus: im ( ε o → 0 ) ⇒ c → ∞⎫⎪ ⎬ If μo is unchanged im ( ε o → ∞ ) ⇒ c → 0 ⎪⎭ 8 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 1 Prof. Steven Errede THE ELECTRIC FIELD E (Vector Quantity!!) (Also known as the Electric Field Intensity) We’ve introduced/discussed the net/time averaged force, F e.g. of Qa acting on Qb: 1 Qa Qb rˆab Fab = 4πε o rab2 We now introduce the concept of a net/time averaged electrostatic field, Ea , due to Qa, at a (separation) distance, rb − ra from Qa (i.e. at Qb), which is defined in terms of the ratio of the net/time averaged force Fab ( rb ) to the strength of the test charge Qb used as a probe: Ea ( rb ) ≡ Fab ( rb ) Qb Fab ( rb ) = Qb Ea ( rb ) or: Fab ẑ Qb • B Point A is known as source point rab Qa • A Qa is known as source charge ra electrostatic force & electrostatic field evaluated rb at point B = “field point” • ŷ O Qb is known as test charge x̂ ra points from the local origin, O to point A where the source charge Qa is located. rb points from the local origin, O to point B where the test charge Qb is located. rb points from the local origin, O to point B where the electric field (net/time averaged) due to Qa is to be evaluated (i.e. by experimentally measuring Fab , and knowing (apriori) Qa and Qb). Fab ( rb ) = Qb Ea ( rb ) = Then: 1 Qa Qb 4πε o r Ea ( rb ) = 2 ab 1 Qa 4πε o r 2 ab rˆab = rˆab = 1 Qa Qb ( rb − ra ) Qa ( rb − ra ) 4πε o rb − ra 3 1 4πε o rb − ra 3 Very often, we will be considering situations in electrostatics where we use one charge, QT to TEST for the presence/existence of “source” charge(s) qs. ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. 9 UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 1 Prof. Steven Errede We want to know e.g. the electric field due to qs, a separation distance, r from it: vector r ≡ ( r − r ′ ) with magnitude: r = r − r ′ Source Point, S ( @ r ′ ) ẑ qs n.b. r r′ r Origin, O ŷ Field Point, P (@ r) QT FT ( r ) = Force on test charge QT (at field point r ), a separation distance r from source charge qs (located at source point, r ′ ): FT ( r ) = 1 QT qs 4πε o r 2 rˆ = 1 QT qs 4πε o r − r ′ 3 ( r − r ′) x̂ primed quantities (e.g. r ′ ) always refer to source (charge) distribution. unprimed quantities (e.g. r ) refer to field/observation point. E (r ) = Electrostatic field ( @ point r ) due to source charge qs a distance r = r − r ′ away from qs: ⎛ 1 ⎞ qs ⎛ 1 ⎞ qs ( r − r ′ ) ⎛ 1 ⎞ qs ( r − r ′ ) ⎛ 1 ⎞ qs E (r ) = ⎜ r − r′) =⎜ =⎜ ⎟ 2 rˆ = ⎜ ⎟ 2 ⎟ ⎟ 2 3 ( ⎝ 4πε o ⎠ r ⎝ 4πε o ⎠ r r − r ′ ⎝ 4πε o ⎠ r − r ′ r − r ′ ⎝ 4πε o ⎠ r − r ′ cumbersome notation, but very explicit!!! FT ( r ) = QT E ( r ) Obviously, SI Units of E ( r ) are Newtons / C (also ≡ volts m) Units of E = force per unit charge (N/C) from dimensional analysis 10 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 1 Prof. Steven Errede A Detail: ⎛ F (r ) ⎞ A more rigorous definition of electric field intensity, E ( r ) is given by: E ( r ) ≡ im ⎜ ⎟⎟ QT → 0 ⎜ Q ⎝ T ⎠ We really do need this limiting process – experimentally/in real life, the presence of a finite-singed test charge QT necessarily perturbs the source charge distribution that one is attempting to measure!! This is especially true for spatially-extended source charge distributions. As the test charge is made smaller and smaller, the perturbing effect on the original/unperturbed source charge distribution is made smaller and smaller. In the limit QT → 0, the true source charge distribution is obtained. THIS IS VERY IMPORTANT TO KEEP THIS IN MIND!!! IT IS NOT A TRIVIAL POINT!!! Usually, we might think of e.g. QT = 1 e and e.g. qs = 1019 e, thus qs >> QT, and thus perturbing effects are negligible (in this case). We have shown that: E ( r ) ≡ F (r ) 1 qs 1 qs QT = rˆ and thus: F ( r ) = rˆ = QT E ( r ) 2 4πε o r 4πε o r 2 QT If F ( r ) is a radial force ⎫⎪ ⎬ for point source charge, qs then E ( r ) is also radial ⎪⎭ Convention: direction of electric field lines for qs = +e and qs = −e • • qs = −e inward qs = +e outward ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. 11 UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 1 Prof. Steven Errede ELECTRIC FIELD LINES Associated with Two Point Charges Equal but opposite charges Figure 2.13 Equal charges Figure 2.14 12 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 1 Prof. Steven Errede THE PRINCIPLE of LINEAR SUPERPOSITION -VERY IMPORTANT⎛ F (r ) ⎞ Assuming we are always in im ⎜ ⎟⎟ (i.e. QT << qs) regime, then suppose we have N discrete point QT → 0 ⎜ Q ⎝ T ⎠ source charges: q1 , q2 , q3 , q4 … qN What is the (total or net) force, FToT ( r ) due to all of the N source charges? N Vectorially, we know that FToT ( r ) = F1 ( r ) + F2 ( r ) + F3 ( r ) + … FN ( r ) = ∑ Fi ( r ) . More explicitly: i =1 N FTOT ( r ) = F1 ( r ) + F2 ( r ) + F3 ( r ) + … EN ( r ) = ∑ Fi ( r ) i =1 ⎧ Q ⎫⎧ q ⎫ Q q q q = ⎨ T ⎬ ⎨ 12 rˆ1 + 22 rˆ2 + 32 rˆ3 + … N2 rˆN ⎬ = T r2 r3 rN ⎩ 4πε o ⎭ ⎩ r1 ⎭ 4πε o where: rˆ ≡ ( r − ri ) = ri N qi rˆ ∑ 2 i i =1 ri What is (total or net) electric field intensity, ETOT ( r ) due to all of the N source charges? We know that: FTOT ( r ) = QT ETOT ( r ) or: ETOT ( r ) ≡ FTOT ( r ) QT N ETOT ( r ) = E1 ( r ) + E2 ( r ) + E3 ( r ) + … EN ( r ) = ∑ Ei ( r ) ∴ i =1 ⎧ 1 ⎫ ⎧ q1 q3 qN ⎫ q2 1 =⎨ ⎬ ⎨ 2 rˆ1 + 2 rˆ2 + 2 rˆ3 + … 2 rˆN ⎬ = r2 r3 rN ⎩ 4πε o ⎭ ⎩ r1 ⎭ 4πε o N qi i =1 i ∑r 2 rˆi We can extend the use of the principle of linear superposition to mathematically describe the net/total force + net/total electric field intensity at the field point, r for arbitrary continuous charge distributions: Q 1 ⎛ 1 ⎞ ⎛ 1 ⎞ FTOT ( r ) = T ∫ ⎜ 2 rˆ ⎟ dqs and ETOT ( r ) = ⎜ rˆ ⎟ dqs 4πε o ⎝ r ⎠ 4πε o ∫ ⎝ r 2 ⎠ where: r = ( r − r ′ ) , r = r − r ′ , rˆ = r r ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. 13 UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 1 Prof. Steven Errede Then for volume, surface & line charge source distributions: A.) VOLUME CHARGE DISTRIBUTIONS: Volume Charge Density, ρ ( r ′ ) : (e.g. inside cylinders, spheres, boxes, etc.) Q ⎛ 1 ⎞ dqs = ρ ( r ′ ) dτ ′ : FTOT ( r ) = T ∫ ⎜ 2 rˆ ⎟ ρ ( r ′ ) dτ ′ 4πε o v ⎝ r ⎠ Coulombs/m3 B.) ETOT ( r ) = 1 4πε o ⎛ 1 ∫ ⎜⎝ r 2 v ⎞ ⎠ rˆ ⎟ ρ ( r ′ ) dτ ′ SURFACE CHARGE DISTRIBUTIONS: Surface Charge Density, σ ( r ′ ) : (e.q. on surfaces of cylinders, spheres, boxes, etc.) Q ⎛ 1 ⎞ dqs = σ ( r ′ ) da′: FTOT ( r ) = T ∫ ⎜ 2 rˆ ⎟ σ ( r ′ ) da′ 4πε o S ⎝ r ⎠ Coulombs/m2 ETOT ( r ) = where: 14 1 4πε o ⎛ 1 ∫ ⎜⎝ r S r = ( r − r′) , 2 ⎞ ⎠ rˆ ⎟ σ ( r ′ ) da′ r = r − r ′ , rˆ = r r ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I C). Fall Semester, 2007 Lecture Notes 1 Prof. Steven Errede LINE CHARGE DISTRIBUTIONS: Linear Charge Density, λ ( r ′ ) : (e.q. wire) dqs = λ ( r ′ ) d ′ : FTOT ( r ) = Coulombs/m where: ETOT ( r ) = QT ⎛ 1 ⎞ ⎜ rˆ ⎟ λ ( r ′ ) d ′ 4πε o C∫ ⎝ r 2 ⎠ 1 4πε 0 r = ( r − r′) , ⎛ 1 ∫ ⎜⎝ r C 2 ⎞ rˆ ⎟ λ ( r ′ ) d ′ ⎠ r = r − r ′ , rˆ = r r Thus, a complete description of all possible charge distributions, consisting of discrete and continuous charge distributions: FTOT ( r ) = QT 4πε o ⎧⎪ N qi ⎛ ρ ( r′) ⎞ ⎛ σ ( r′) ⎞ ⎛ λ ( r ′) ⎞ ⎨∑ 2 rˆi + ∫ ⎜ 2 rˆ ⎟ dτ ′ + ∫ ⎜ 2 rˆ ⎟ da′ + ∫ ⎜ 2 rˆ ⎟ d r r r ⎪⎩ i =1 r i ⎠ ⎠ ⎠ V⎝ S⎝ C⎝ ⎫⎪ ′⎬ ⎪⎭ ETOT ( r ) = ⎛ ρ ( r′) ⎞ ⎛ σ ( r ′) ⎞ ⎛ λ ( r′) ⎞ 1 ⎪⎧ N qi ⎨∑ 2 rˆi + ∫ ⎜ 2 rˆ ⎟ dτ ′ + ∫ ⎜ 2 rˆ ⎟ da′ + ∫ ⎜ 2 rˆ ⎟ d r r r 4πε o ⎩⎪ i =1 r i ⎠ ⎠ ⎠ V⎝ S⎝ C⎝ ⎪⎫ ′⎬ ⎭⎪ Please Note: For all integrals (above), when integrals over dτ ′ , da′, and/or d ′ are carried out, FTOT ( r ) and thus ETOT ( r ) have NO r ′ (i.e. source-position) dependence - it has been integrated over/integrated out!!! FTOT ( r ) and ETOT ( r ) ≡ FTOT ( r ) QT are functions of the field point variable r ONLY i.e. they are not functions of r ′ (source point{s}) !!! PLEASE work/grind through example 2.1 Griffiths p. 62-63) on your own to better learn/understand this! ACTIVE “LEARNING BY DOING” ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. 15 UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 1 Prof. Steven Errede EXAMPLE 2.1 p. 62 Griffiths - very explicit detailed derivationFind the electric field intensity, E(r) a distance z above mid-point of a straight line segment of length 2L, which carries a uniform line charge λ (Coulombs/meter) (n.b. QTOT = 2λ L ) λ ( r ′) 1 Here, E ( r = zzˆ ) = rˆd ′ 4πε o C∫ r 2 Notice the symmetry of this problem – contribution to net E ( r ) @ field point, P from infinitesimal line charges d λ associated with infinitesimal line segments, dL located at ± x such that x̂ components of net electric field @ field point, P cancel each other: cos θ = sin θ = z r x r = = z x2 + z 2 x x2 + z 2 dENET ( r = zzˆ ) = dE + ( r = zzˆ ) + dE − ( r = zzˆ ) ⎧⎪⎛ 1 = ⎨⎜ ⎪⎩⎝ 4πε o ⎫⎪ ⎞ ⎛ λ dL ⎞ + ⎟ ⎜ 2 ⎟ ⎡⎣( − sin θ xˆ ) + ( cos θ zˆ ) ⎤⎦ ⎬ ← dE ( r = zzˆ ) ⎪⎭ ⎠⎝ r ⎠ ⎧⎪⎛ 1 + ⎨⎜ ⎪⎩⎝ 4πε o ⎫⎪ ⎞ ⎛ λ dL ⎞ − ⎟ ⎜ 2 ⎟ ⎡⎣( + sin θ xˆ ) + ( cos θ zˆ ) ⎤⎦ ⎬ ← dE ( r = zzˆ ) ⎪⎭ ⎠⎝ r ⎠ ⎛ 1 ⎞ ⎛ λ dL ⎞ =2 ⎜ ⎟ ⎜ 2 ⎟ cos θ zˆ 4 πε ⎠ o ⎠⎝ r ⎝ 16 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 1 Prof. Steven Errede Now only need to integrate this expression over x from 0 ≤ x ≤ L : { } L L ⎛ 1 + − ˆ ˆ = + = = dE r zz dE r zz ( ) ( ) ∫0 ∫0 ∫0 2 ⎜⎝ 4πε o 0 ⎤⎡ L⎡ ⎤ ⎛ 1 ⎞ 1 z ⎢ ⎥⎢ = 2⎜ λ ⎟ ∫0 ⎥ dx zˆ 2 2 ⎢⎣ ( x + z ) ⎥⎦ ⎣ x 2 + z 2 ⎦ ⎝ 4πε o ⎠ ENET ( r = zzˆ ) = ∫ dE ( r = zzˆ ) = L L ⎞ ⎛ λ dL ⎞ ⎟ ⎜ 2 ⎟ cos θ zˆ ⎠⎝ r ⎠ ⎛ 2λ z ⎞ L 1 =⎜ dx zˆ ⎟ ∫0 2 2 3/ 2 4 πε o ⎠ ⎝ (x + z ) ⎤ ⎛ 2λ z ⎞ ⎡ x =⎜ ⎟⎢ 2 2 ⎥ 2 ⎝ 4πε o ⎠ ⎣ z x + z ⎦ L 0 ⎛ 2λ ⎞ ⎡ z zˆ = ⎜ ⎟⎢ 2 ⎝ 4πε o ⎠ ⎣ z ⎤ ⎞ ⎛ 2λ L ⎞ ⎛ 1 ⎟ zˆ ⎟⎜ ⎥ zˆ = ⎜ 2 2 L +z ⎦ ⎝ 4πε o ⎠ ⎝ z z + L ⎠ L 2 2 2 ⎛ ε⎞ ⎛L⎞ If z >> L; then (Taylor Series Expansion) 1 + ε ≈ ⎜1 + ⎟ ≈ 1 for ε = ⎜ ⎟ 1 ⎝ 2⎠ ⎝z⎠ Q 2λ L ENET ( r = zzˆ ) ≈ zˆ = TOT 2 zˆ ← same E-field as that due to a point charge, q! 2 4πε o z 4πε o z If L → ∞ (i.e. infinite straight wire): use the same Taylor series expansion, but for L z : ⎛ ε⎞ ⎛z⎞ i.e. 1 + ε ≈ ⎜1 + ⎟ ≈ 1 for ε = ⎜ ⎟ ⎝ 2⎠ ⎝L⎠ Then: ENET ( r = zzˆ ) ≈ 2 1 2λ λ zˆ = zˆ 4πε o z 2πε o z 1 The E-field is actually in the radial ( ρ̂ ) direction for an infinite straight wire – in cylindrical coordinates: ENET ( r ) ≈ 1 2λ 4πε o ρ ρˆ = λ ρˆ 2πε o ρ ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. 17