AP-C Electric Force and Electric Field AP-C Objectives (from College Board Learning Objectives for AP Physics) 1. Charge and Coulomb’s Law a. Describe the type of charge and the attraction and repulsion of charges b. Describe polarization and induced charges. c. Calculate the magnitude and direction of the force on a positive or negative charge due to other specified point charges. d. Analyze the motion of a particle of specified charge and mass under the influence of an electrostatic force. e. Describe the process of charging by induction. f. Explain why a neutral conductor is attracted to a charged object. 2. Electric Field due to Point Charges a. Define the electric field in terms of force on a test charge. b. Describe and calculate the electric field produced by one or more point charges. c. Calculate the magnitude and direction of the force on a positive or negative charge placed in a specified field. d. Interpret electric field diagrams. 3. Gauss’s Law a. Understand the relationship between electric field and electric flux. i. Calculate the flux of an electric field through an arbitrary surface or of a uniform field over and perpendicular to a Gaussian surface. ii. Calculate the flux of the electric field through a rectangle when the field is perpendicular to the rectangle and a function of one coordinate only. iii. State and apply the relationship between flux and lines of force. b. Understand Gauss’s Law i. State Gauss’s Law in integral form and apply it qualitatively to relate flux and electric charge for a specified surface. ii. Apply Gauss’s Law, along with symmetry arguments, to determine the electric field for a planar, spherical, or cylindrically symmetric charge distribution. iii. Apply Gauss’s Law to determine the charge density or total charge on a surface in terms of the electric field near the surface. 4. Electric Fields due to Other Charge Distributions a. Calculate the electric field of a straight, uniformly charged wire; the axis of a thin ring of charge; and the center of a circular arc of charge. b. Identify situations in which the direction of the electric field produced by a charge distribution can be deduced from symmetry considerations. c. Describe the patterns and variation with distance of the electric field of oppositely charged parallel plates; a long uniformly charged wire; a thin cylindrical shell; and a thin spherical shell. d. Determine the fields of parallel charged planes, coaxial cylinders, and concentric spheres. - 1 - Charge and Coulomb's Law Electric Charge AP-C Objectives (from College Board Learning Objectives for AP Physics) Electric charge (q) is a fundamental property of certain particles. The smallest 1. Charge and Coulomb’s Law amount of isolatable charge is the elementary charge (e), equal to 1.6×10-19 a. Describe the type of charge and the attraction and repulsion of charges coulombs. Charge can be positive or negative. b. Describe polarization and induced charges. c. Calculate the magnitude and direction of the force on a positive or negative charge due to other specified point charges. d. Analyze the motion of a particle of specified charge and mass under the influence of an electrostatic force. Atomic Particles Conductors and Insulators e. Explain the process of charging by induction. protons have a charge of +1e Charges can move freely in conductors. electrons have a charge of −1e Charges cannot move freely in insulators. f. Explain why a neutral conductor is attracted to a charged object. neutrons are neutral 2. Electric Field due to Point Charges Atoms with an excess of protons a. Define the electric field in terms of force on a test charge. or electrons are known as ions. b. Describe and calculate the electric field produced by one or more point charges. c. Calculate the magnitude and direction of the force on a positive or negative charge placed in a specified field. Polarization and Electric Dipole Moment d. Interpret electric field diagrams. When a charged object is brought near a conductor, the electrons in the conductor are free to move. When a charged object is brought near an insulator, the electrons are not free to move, but they may spend a little more time on one side of their orbit than another, creating a net separation of charge in a process known as Electric Field (E) Electric Force and Coulomb’s Law polarization. The distance between the shifted positive and negative charges, The electric field describes the amount of electrostatic multiplied by the charge, is known as the electric dipole moment. Like charges repel, opposite charges attract. force observed by a charge placed at a point in the conductor insulator field per unit charge. The electric field vector points in 2 1 q q −12 - 1 2 C + - + - + the direction a positive test charge would feel a force. F = ε = 8.85×10 2 + 2 0 + - + - + Electric field strength is measured in N/C, which are N i m + + 4πε0 r + - + + - + equivalent to V/m. + - + + - + - + - + F 1 q + + + - + - + E F qE E charged rod - charged+ rod = = = 2 + + + - + + + q 4πε0 r + + + + Electric field lines indicate the direction of the electric force on a positive test charge. Conduction and Induction Charging by contact is known as conduction. If a charged conductor is brought into contact with an identical neutral conductor, the net charge will be shared across the two conductors. Charging an object without placing it in contact with another charged object is known as induction. Electric Field due to Multiple Point Charges The electric field follows the law of superposition. In order to determine the electric field due to multiple point charges, add up the electric field due to each of the individual charges. Sample Problem: E Field due to Point Charges Find the electric field at the origin due to the three point charges shown. y (m) 1 q 1 (2) 8 E1 = 2 = 2 =< 0,−2.81×10 > 9 4πε0 r 4πε0 8 8 +2C 1 q 1 (1) 9 N E = = = 1.13×10 C → 7 2 2 2 2 2 4πε0 r 4πε0 ( 2 + 2 ) E E E E 6 9 N 9 N tot = 1 + 2 + 3 → E2 =< −1.13×10 C × cos45°,−1.13×10 C × sin45° >→ 5 8 9 8 8 N E =< −7.95×10 ,−7.95×10 > N Etot =< −5.14 ×10 ,−1.08 ×10 > C 4 2 C 1 q 1 2 3 E 2.81 108 ,0 N +1C 3 = 2 = 2 =< × > C 2 4πε0 r 4πε0 8 1 -2C x (m) 1 2 3 4 5 6 7 8 9 Sample Problem: Where is the E Field Zero? Determine the x-coordinate where the electric field is zero using the diagram. 1 q E = 1 1 4πε r 2 r 11-r 0 +1C +2C x (m) 1 q2 E = 2 4 (11 r)2 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 πε0 − 1 q k= 1 1 1 q2 4πε0 k(1) k(2) 2 Etot = E1 + E2 = 2 − 2 ⎯⎯⎯→ 2 − 2 = 0 → r + 22r −121= 0 → r = 4.56 4πε0 r 4πε0 (11− r) r (11− r) x = −6 + 4.56 = −1.44 - 2 - Electric Fields due to Other Charge Distributions 1 Charge Densities AP-C Objectives (from College Board Learning Objectives for AP Physics) ΔQ 1. Electric Fields due to Continuous Charge Distributions Linear Charge Density λ = a. Calculate the electric field of a straight, uniformly charged wire; the axis of a thin ring of ΔL charge; and the center of a circular arc of charge. b. Identify situations in which the direction of the electric field produced by a charge distribution ΔQ can be deduced from symmetry considerations. Surface Charge Density σ = ΔA ΔQ Volume Charge Density ρ = ΔV E-Field Due to a Thin Uniform Semicircle of Charge A thin insulating semicircle of charge Q with radius R is centered around point C. Determine the electric field at point C due to the semicircle of charge. Symmetry Arguments Horizontal component will cancel out since the charge is uniformly distributed, so we only need to worry about the vertical component of the electric field. 1 dQ θ=π 1 λRdθ ΔQ Q dE = sinθ ⎯d⎯Q=λ⎯Rdθ⎯→ dE = sinθ dθ → λ = = y 2 ∫ y ∫θ=0 2 ΔL πR 4πε0 R 4πε0 R λ θ=π λ π 2λ λ E = sinθ dθ → E = (−cosθ) = = y ∫θ=0 y 0 4πε0 R 4πε0 R 4πε0 R 2πε0 R E Field Due to a Thin Straight Insulating Wire Find the electric field some distance d from a long straight insulating rod of length L at Q λ = a point P which is perpendicular to the wire and equidistant from each end of the wire. L Strategy 1. Divide the total charge Q into smaller charges ΔQ. 2. Find the electric field due to each ΔQ. 3. Find the total electric field by adding up the individual electric fields due to each ΔQ. 4. Realize the y-component of the electric field is 0 due to symmetry arguments. 2 2 1 ΔQ ri = yi +d Ei = Ei cosθi = 2 cosθi ⎯⎯⎯d ⎯d⎯⎯→ x cosθ = = 4πε r i 2 2 0 i ri yi +d Q Q dy 1 ΔQ d 1 dΔQ Δ = L Ei = 2 2 = 3 ⎯⎯⎯⎯→ x 4πε ( y + d ) 2 2 4πε 2 2 2 0 i yi + d 0 ( yi + d ) y L 1 dQdy = 2 dQ / L dy E E i = 3 → x = L 3 → x 2 2 y=− 2 2 4πε 2 ∫ 2 4πε 2 0 L( yi + d ) 0 ( y + d ) dx x = +C L ∫ 3 2 2 2 dQ / L 2 dy 2 2 2 a a +x E (a +x ) x = L 3 ⎯⎯⎯⎯⎯⎯⎯⎯→ ∫− 2 2 2 4πε0 2 ( y + d ) L ⎛ 2 ⎞ dQ / L y Q / L ⎡ L / 2 −L / 2 ⎤ Ex = ⎜ 1 ⎟ = ⎢ 1 − 1 ⎥ → 4πε 2 2 2 2 4πε d L 2 2 2 − L 2 2 2 0 ⎜ d ( y + d ) L ⎟ 0 ⎢(( 2 ) + d ) (( 2 ) + d ) ⎥ ⎝ − 2 ⎠ ⎣ ⎦ Q / L L Q Ex = 1 = 4πε d L 2 2 2 L 2 2 0 (( 2 ) + d ) 4πε0d ( 2 ) + d What is the E Field if the rod is infinite? Q ⎛ ⎞ λ= Q Q Q L λ Ex = lim⎜ ⎟ = = ⎯⎯⎯→ Ex = L→∞ ⎜ L 2 2 ⎟ 4πε d( L ) 2πε dL 2πε d ⎝ 4πε0d ( 2 ) + d ⎠ 0 2 0 0 Note that Q approaches infinity as L approaches infinity! What is the E field if the distance d is infinite? ⎛ ⎞ Q Q ⎜ ⎟ Acts like the E-field of a point charge! Ex = lim = 2 d→∞ ⎜ 2 2 ⎟ 4 d 4 d L d πε0 ⎝ πε0 ( 2 ) + ⎠ - 3 - Electric Fields due to Other Charge Distributions 2 Charge Densities AP-C Objectives (from College Board Learning Objectives for AP Physics) ΔQ 1.