Lecture 2 - Coulomb’s Law and Electric Field Chapters 25 and 26 - Thursday January 11th •Quick review of Tuesday’s class •Coulomb’s Law •Scalar and vector notation •Discrete charge distributions (superposition principle) •Continuous charge distributions •Electric fields •Discrete charge distributions •Continuous charge distributions •Electric field lines Reading: pages 567 thru 597 (chapters 25 & 26) in HRK Read and understand the sample problems WebAssign: set 1, due Thur. 18th at 11:59pm Graded problems: Ch. 26 – Ex. 8, 13, 18, 36; Prob. 10 Practice problems: Ch. 26 - Ex. 7, 14, 23 ; Prob. 1 What is charge? •Charge is measured in Coulomb’s (C) •Fundamental unit. •Definition based on forces between current carrying wires (current = Ampères, or C/s), i.e. chapter 33. •Charge is discrete •Thompson discovered the electron in 1896. He found that charge was carried by elementary particles with the same charge to mass ratio. •The elementary charge of the electron was not measured until 1909 (Millikan). •Both experiments earned Nobel prizes. Charge on an electron: 1 Coulomb of charge: 1 Ampère (= 1 C/s) Charge is discrete: e = 1.6 × 10−19 Coulombs 6.24 × 1018 electrons 6.24 × 1018 electrons/second q = ne n = ±1, ±2, ±3,... Static electricity through charging Conductor Insulator Coulomb’s Law Coulomb’s torsional balance 1785 q1 q2 F∝ , 2 r K= 1 4πε o or q1 q2 F=K r2 = 8.99 × 109 N ⋅ m 2 / C2 εο = 8.85418781762 × 10-12 C2/N·m2 1 εo = −7 2 2 2 4 10 N s / C π × ⋅ × c ( ) c = speed of light in vacuum = 299792458 m/s Coulomb’s Law in vector notation G F12 = G 1 q1q2 rˆ12 = − F21 2 4πε o r12 G F21 = 1 q1q2 rˆ21 2 4πε o r21 G r12 rˆ12 = = −rˆ21 r12 Superposition principle Source charges Test charge q0 G G G G G G Fq0 = F1 + F2 + F3 + ...Fi = ∑ Fi i •Leads to Maxwell’s equations being linear. Calculus (continuous charge distribution): G G F = ∫ dF = iˆ ∫ dFx + ˆj ∫ dFy + kˆ ∫ dFz G dF = q0 dq rˆ 2 4πε o r 1 q0 = test charge Charge densities In 1D (a line or wire): Q λ= , L or dQ λ= dL λ is the line charge density, or charge per unit length, in Coulombs per meter. L represents length, and Q is charge. In 2D (a surface or sheet): σ = Q , A or dQ σ= dA σ is the surface charge density, or charge per unit area in Coulombs per meter2; A represents area, and Q is charge. In 3D (a solid object): Q ρ= , V or ρ is the volume charge density, or charge per unit volume in Coulombs per meter3. V represents volume, and Q is charge. dQ ρ= dV Electric field - Chapter 26 G GMm m Analogy with gravitation F = − 2 rˆ r •Problem: Force depends on test mass! •But: G G G G F F = mg or g = m G GM g = − 2 rˆ r • Same for all masses. • Represents the gravitational influence (field) of the mass M. Electric field - Chapter 26 G qq q0 + Analogy with gravitation F = K 20 rˆ r •Problem: Force depends on test charge! G G G G F F = q0E or E = q0 •Definition: G E= 1 q rˆ 2 4πε o r • Same for all test charges. • Represents the electrostatic influence (field) of the charge q. source charge q = −Q Electric field - Chapter 26 Newton’s law for electrostatics: G G F = q0E There’s really no need for the “test charge” G G F = qE G This is the force on a charge q in an electric field E Units for E are N/C in this chapter (later we shall use volts per meter) Electric field - Chapter 26 Coulomb’s law: G E= 1 q ˆ r 2 4πε o r •Note that the force may be directed both towards and away from the source charge, since this charge may be either positive or negative. G •Length of arrow signifies magnitude of E Of course, superposition principle still holds: G G G G G G E = E1 + E2 + E3 + ...Ei = ∑ Ei i G G E = ∫ dE = iˆ ∫ dE x + ˆj ∫ dE y + kˆ ∫ dE z Electric field lines • The tangent to an electric field line at a point in space gives the direction of the electric field at that point. • The magnitude of the electric field at any point is proportional to the number of field lines per unit crosssectional area perpendicular to the lines. Electric field lines • Electric field lines start on positive charges and end on negative charges (can also start/end at infinity). • The symmetry of the problem dictates the directions in which field lines radiate from charges. Electric field lines • The number of field lines radiating from a charge is proportional to the charge.