Northeastern Illinois University Electricity & Magnetism I Electrostatics: Basic Principles I Greg Anderson Department of Physics & Astronomy Northeastern Illinois University Spring 2017 c 2004-2017 G. Anderson Electricity & Magnetism – slide 1 / 71 Northeastern Illinois Overview University Electric Charge Coulomb’s Law The Electric Field Examples Curl, Divergence & Potential Gauss’s Law Dirac Delta Green Funct. Potential Multipole Torque and Potential c 2004-2017 G. Anderson Electricity & Magnetism – slide 2 / 71 Northeastern Illinois University Electric Charge Electromagnetism Quantization Quantization II Quantization III Conservation Conservation II Coulomb’s Law The Electric Field Electric Charge Examples Curl, Divergence & Potential Gauss’s Law Dirac Delta Green Funct. Potential Multipole Torque and Potential c 2004-2017 G. Anderson Electricity & Magnetism – slide 3 / 71 Northeastern Illinois Electromagnetism University With the exception gravity, almost every force that you experience in everyday life is electro-magnetic in origin. ◆ EM forces bind electrons and nuclei into atoms. ◆ EM forces bind atoms into molecules. O H H ◆ EM forces bind atoms & molecules into solids. Electric forces are produced by electric charges. c 2004-2017 G. Anderson Electricity & Magnetism – slide 4 / 71 Northeastern Illinois Electric Charge Quantization University Electric charge is quantized: A charged object has a surplus or deficit in the number of electrons relative to protons. ◆ e = fundamental unit of charge Q =+e, Q = e proton electron − ◆ For any charge: Q = ne, n =0, 1, 2,... ± ± SI Units: [Charge] = Coulomb = C e =1.6 10−19 C. × c 2004-2017 G. Anderson Electricity & Magnetism – slide 5 / 71 Northeastern Illinois Electric Charge Quantization II University Electric charge is quantized: A charged object has a surplus or deficit in the number of electrons relative to protons. For any charged object: Q = ne, n =0, 1, 2,... ± ± ◆ Let Ne be the number of electrons in an object ◆ Let Np be the number of protons in an object. n = N N p − e c 2004-2017 G. Anderson Electricity & Magnetism – slide 6 / 71 Northeastern Illinois Charge & Charged Constituents University The total charge of any composite object is the sum of the charges of its charged constituents: electrons and protons. ◆ N electrons, each with charge e contribute: e − Q = N ( e)= N e electrons e − − e ◆ Np protons, each with charge +e contribute: Qprotons = Npe Example + − N = 3 Together they give a total charge: + p +− − Ne = 4 Q =(Np Ne)e − Q = 1e − − c 2004-2017 G. Anderson Electricity & Magnetism – slide 7 / 71 Northeastern Illinois Electric Charged is Conserved University Electric Charge is Conserved: The total charge of an isolated system never changes. For an Isolated System: Qinitial = Qfinal c 2004-2017 G. Anderson Electricity & Magnetism – slide 8 / 71 Northeastern Illinois Charge Conservation Examples University Examples of electric charge conservation: Pair production − − γ + γ e+ + e + −→ + Pair annihilation e+ + e− γ + γ −→ − Ionization H − − H + γ p+ + e + −→ c 2004-2017 G. Anderson Electricity & Magnetism – slide 9 / 71 Northeastern Illinois University Electric Charge Coulomb’s Law Electrostatics Selected E&M Heros Coulomb’s Law Gravitational Analogy Coulomb’s Law II Coulomb’s Law & Superposition Coulomb’s Law The Electric Field Examples Curl, Divergence & Potential Gauss’s Law Dirac Delta Green Funct. Potential Multipole Torque and Potential c 2004-2017 G. Anderson Electricity & Magnetism – slide 10 / 71 Northeastern Illinois Electrostatics University Electrostatics: The science of the interactions between electric fields and electric charges in static configurations. All of electrostatics can be reduced to: ◮ Coulomb’s Law ◮ Principle of superposition. c 2004-2017 G. Anderson Electricity & Magnetism – slide 11 / 71 Northeastern Illinois Selected E&M Heros University Volta Amp´ere Oersted Coulomb Stokes Laplace Maxwell Galvani Thomson (Kelvin) Franklin Faraday DuFey Gauss 1700 1750 1800 1850 1900 1950 c 2004-2017 G. Anderson Electricity & Magnetism – slide 12 / 71 Northeastern Illinois Coulomb’s Law University Charles Augustin de Coulomb (1785) y q Empirical force of Q on q: qQ qQ F r r q = K 2 ˆ = 2 ˆ r r 4πǫ0r ˆr = unit vector from Q to q ˆr r ˆr = x r Q | | 1 K = =8.99 109 N m2/C2 4πǫ0 × · [q] = Coulomb = C, ǫ0 = permittivity of the vacuum. c 2004-2017 G. Anderson Electricity & Magnetism – slide 13 / 71 Northeastern Illinois Gravitational Analogy University m q r r ˆr ˆr M Q Mm Qq F = G ˆr F = K ˆr − r2 r2 c 2004-2017 G. Anderson Electricity & Magnetism – slide 14 / 71 Northeastern Illinois Coulomb’s Law II University A more general notation: Electrostatic force of q2 on q1 q1 q q r F = K 1 2 ˆr, ˆr = 1 r2 r r | | x1 ˆr x + r = x , r = x x , 2 1 ⇒ 1 − 2 q2 q1q2 x1 x2 x2 F = − on q 1 4πǫ x x 3 1 0 | 1 − 2 | 1 What is the force on q2? K = 4πǫ0 F2 = c 2004-2017 G. Anderson Electricity & Magnetism – slide 15 / 71 Northeastern Illinois Coulomb’s Law II University A more general notation: Electrostatic force of q2 on q1 q1 q q r F = K 1 2 ˆr, ˆr = 1 r2 r r | | x1 ˆr x + r = x , r = x x , 2 1 ⇒ 1 − 2 q2 q1q2 x1 x2 x2 F = − on q 1 4πǫ x x 3 1 0 | 1 − 2 | 1 q1q2 x2 x1 K = F2 = − on q2 4πǫ0 4πǫ x x 3 0 | 1 − 2 | c 2004-2017 G. Anderson Electricity & Magnetism – slide 15 / 71 Northeastern Illinois Coulomb’s Law & Superposition University Force on q from q q i j r1 q1 q q x x F = i j i − j r2 i 4πǫ x x 3 0 | i − j | r x1 3 q2 x Force on q from q1, q2, . , qN x2 N q qqk x xk x3 3 F q = − 3 4πǫ0 x xk Xk=1 | − | Electric field, force per unit charge: F N q x x E x q k k ( ) = lim = − 3 q→0 q 4πǫ0 x xk Xk=1 | − | c 2004-2017 G. Anderson Electricity & Magnetism – slide 16 / 71 Northeastern Illinois University Electric Charge Coulomb’s Law The Electric Field Action at a Distance Field Theory Electric Monopole Field + Electric Monopole Field The Electric Field − The Electric Field Example: Hexagon Example: Hexagon II Electric Field & Charge Continuum Charge Densities Examples Curl, Divergence & Potential Gauss’s Law Dirac Delta Greenc 2004-2017 Funct. G. Anderson Electricity & Magnetism – slide 17 / 71 Northeastern Illinois Action at a Distance? University Coulomb’s law appears to be a theory of “action at a distance.” F + Q Q F = k 1 2 r2 + F Q: What is the mediator of the interaction between these spacially separated objects? A: Electromagnetic Fields. c 2004-2017 G. Anderson Electricity & Magnetism – slide 18 / 71 x Northeastern Illinois Field Theory University Action at a Distance: q q x x F = i j i − j j on i 4πǫ x x 3 0 | i − j | Field Theory (Maxwell and Faraday): ◆ Charges produce fields: E F N q x x E x q k k ( ) = lim = − 3 q→0 q 4πǫ0 x xk Xk=1 | − | ◆ Forces are local interactions between charges and fields: F = qE c 2004-2017 G. Anderson Electricity & Magnetism – slide 19 / 71 Northeastern Illinois Electric Monopole Field + University + c 2004-2017 G. Anderson Electricity & Magnetism – slide 20 / 71 Northeastern Illinois Electric Monopole Field + University + c 2004-2017 G. Anderson Electricity & Magnetism – slide 20 / 71 Northeastern Illinois Electric Monopole Field + University + c 2004-2017 G. Anderson Electricity & Magnetism – slide 20 / 71 Northeastern Illinois Electric Monopole Field University − − c 2004-2017 G. Anderson Electricity & Magnetism – slide 21 / 71 Northeastern Illinois Electric Monopole Field University − − c 2004-2017 G. Anderson Electricity & Magnetism – slide 21 / 71 Northeastern Illinois Electric Monopole Field University − − c 2004-2017 G. Anderson Electricity & Magnetism – slide 21 / 71 Northeastern Illinois The Electric Field University Electric field from q P 1 r q1 1 q1 x x1 E1(x)= − r2 4πǫ x x 3 0 | − 1 | r x1 3 q2 x Electric field from q1, q2, . , qN x2 N q qk x xk x3 3 E x ( )= − 3 4πǫ0 x xk Xk=1 | − | Electric force on a test charge q: F = qE c 2004-2017 G. Anderson Electricity & Magnetism – slide 22 / 71 Northeastern Illinois Charges on a Regular Hexagon University Electric field at x y 6 ′ 2 1 q x x E x k k ( )= − ′ 3 4πǫ0 x x a k=1 | − k | π X 3 Position of charge k 3 6 x kπ kπ x′ = ˆia cos + ˆja sin k 3 3 4 5 1 √3 1 √3 ( , ):( 1, 0) : ( , ) ± 2 2 ± ± 2 − 2 q 6 (x a cos kπ/3)ˆi +(y a sin kπ/3)ˆj E x ( )= − − 3/2 4πǫ0 [(x a cos kπ/3)2 +(y a sin kπ/3)2] Xk=1 − − c 2004-2017 G. Anderson Electricity & Magnetism – slide 23 / 71 Northeastern Illinois Charges on a Regular Hexagon II University y 2 1 6 ′ a qk x xk π E(x)= − ′ 3 x x 3 3 6 x 4πǫ0 k Xk=1 | − | 1 √3 1 √3 ( , ):( 1, 0) : ( , ) ± 2 2 ± ± 2 − 2 4 5 q 6 (x a cos kπ/3)ˆi +(y a sin kπ/3)ˆj E x ( )= − − 3/2 4πǫ0 [(x a cos kπ/3)2 +(y a sin kπ/3)2] Xk=1 − − On the x axis (y = 0): q 6 (x−a cos kπ/3) Ex(x, 0) = 3/2 4πǫ0 k=1 [(x−a cos kπ/3)2+(a sin kπ/3)2] −aq 6 (sin kπ/3) Ey(x, 0) = P 3/2 =0 4πǫ0 k=1 [(x−a cos kπ/3)2+(a sin kπ/3)2] P c 2004-2017 G. Anderson Electricity & Magnetism – slide 24 / 71 Northeastern Illinois Charges on a Regular Hexagon II University y 2 1 6 ′ a qk x xk π E(x)= − ′ 3 x x 3 3 6 x 4πǫ0 k Xk=1 | − | 1 √3 1 √3 ( , ):( 1, 0) : ( , ) ± 2 2 ± ± 2 − 2 4 5 On the x axis (y = 0): q 6 (x−a cos kπ/3) Ex(x, 0) = 3/2 4πǫ0 k=1 [(x−a cos kπ/3)2+(a sin kπ/3)2] −aq 6 (sin kπ/3) Ey(x, 0) = P 3/2 =0 4πǫ0 k=1 [(x−a cos kπ/3)2+(a sin kπ/3)2] P Far from the origin (x a) (1+ δ)−3/2 = 1 3 δ + 15 δ2 + ≫ − 2 8 ··· 1 6q 9qa2 E (x, 0) + x ≈ 4πǫ x2 2x4 0 c 2004-2017 G.