Electricity and Magnetism Review 1: Units 1-6

Review Formulas Coulomb’s Law Force law between point charges q1 q2 Electric Field Force per unit charge Property of Space Created by Charges Superposition Gauss’ Law Flux through closed surface is always proportional to charge enclosed Gauss’ Law Can be used to determine E field Spheres Cylinders Infinite Planes Electric Potential Potential energy per unit charge Electric Potential Scalar Function that can be used to determine E 2

Applications for Conductors Charges free to move What Determines How They Move? They move until E = 0 ! E = 0 in conductor determines charge densities on surfaces Spheres Cylinders Infinite Planes Gauss’ Law Field Lines & Equipotentials Field Lines Equipotentials Work Done By E Field Change in Potential Energy 3

Example Consider two point charges q1 and q2 located as shown. 1. Find the resultant electric field due to q1 and q2 at the location of q3. 2. Find the resultant force on q3. 3. Find the electric potential due to q1 and q2 at the location of q3. Use components F = 9e9 x (1.6e-19 * 3e22)^2 / 100 = 2 e 15 Newtons Equivalent to weight of 2e14 kg object near surface of the earth paperclip 1e-3 kg textbook 1 kg person 200 kg car 2000 kg Aircraft carrier 97000 tons = 1e5 x 1e3 = 1e 8 kg Mt Everest 9000 m height,. Estimate volume as 1/3 h^3 = 1/3 * (9e3)^3 = 243 e 9 m^3. (if density = 1000 kg/m^3) = 2.43e14 kg.

Example: Spherical Symmetry A solid insulating sphere of radius R has uniform charge density ρ and carries total charge Q. Find the Electric field everywhere. R x y Q r Choose a suitable Gaussian surface: A sphere Calculate the charge enclosed within the Gaussian surface for r > R and for r < R For r > R: For r < R:

Example A solid insulating sphere of radius R has uniform volume charge density ρ and carries total charge Q. Find the Potential difference between two points inside the sphere A and B at distances rA and rB . R x y Q From the previous problem we know that for r < R:

Example: Cylindrical Symmetry Find the electric field at a distance r from a line of positive charge of infinite length and constant linear charge density λ. Choose suitable Gaussian surface: A cylinder Calculate the charge enclosed within the Gaussian surface

Example: Planar Symmetry Find the electric field at a distance r due to an infinite plane of positive charge with uniform surface charge density σ. Choose suitable Gaussian surface: A cylinder perpendicular to the plane Calculate the charge enclosed within the Gaussian surface

Example Point charge +3Q at center of neutral conducting shell of inner radius r1 and outer radius r2. a) What is E everywhere? Use Gaussian surface = sphere centered on origin neutral conductor r1 r2 y x +3Q r < r1 r > r2 r1 < r < r2

Example Point charge +3Q at center of neutral conducting shell of inner radius r1 and outer radius r2. What is E everywhere? b) What is charge distribution at r1? neutral conductor r1 r2 y x +3Q r1 r2 +3Q r1 < r < r2

Example Suppose we give the conductor a charge of -Q a) What is E everywhere? b) What are charge distributions at r1 and r2? r1 r2 -3Q + +2Q +3Q r1 < r < r2 r < r1 r > r2

Example Charge q1 = 2μC is located at the origin. Charge q2 = - 6μC is located at (0, 3) m. Charge q3 = 3.00 μC is located at (4, 0) m   Find the total energy required to bring these charges to these locations starting from infinity.

Example Point charge q at center of concentric conducting spherical shell of radii a1, and a2. The shell carries charge Q. What is V as a function of r? cross-section Charges q and Q will create an E field throughout space metal +Q a1 a2 +q 1. Spherical symmetry: Use Gauss’ Law to calculate E everywhere 2. Integrate E to get V The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class.

Example Gauss’ law: r > a2: +Q a1 < r < a2 : +q r < a2 r cross-section metal +Q a1 a2 r +q r > a2: a1 < r < a2 : r < a2 The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class.

Example To find V: 1) Choose r0 such that V(r0) = 0, usual: r0 = ∞ +Q cross-section To find V: 1) Choose r0 such that V(r0) = 0, usual: r0 = ∞ 2)Integrate! a4 a3 +Q +q r > a2 : a1 < r < a2 : metal r < a1 : The purpose of this Check is to jog the students minds back to when they studied work and potential energy in their intro mechanics class.

Example A rod of length ℓ has a total charge Q uniformly distributed. Find V at point P.

Example Charge is uniformly distributed along the x-axis from the origin to x = a. The charge density is l C/m. What is the x-component of the electric field at point P: (x,y) = (a,h)? y P x a h r dq = l dx What is ?

Example Charge is uniformly distributed along the x-axis from the origin to x = a. The charge density is l C/m. What is the x-component of the electric field at point P: (x,y) = (a,h)? x a P r q1 q dq = l dx h y

Example Charge is uniformly distributed along the x-axis from the origin to x = a. The charge density is l C/m. What is the x-component of the electric field at point P: (x,y) = (a,h)? x a P r q1 q dq = l dx h y

Example The spheres have the same known mass m and charge q and are in equilibrium. Given the angle θ and the length L, find the charge q on each sphere. Is equilibrium possible if the charges are different? F = 9e9 x (1.6e-19 * 3e22)^2 / 100 = 2 e 15 Newtons Equivalent to weight of 2e14 kg object near surface of the earth paperclip 1e-3 kg textbook 1 kg person 200 kg car 2000 kg Aircraft carrier 97000 tons = 1e5 x 1e3 = 1e 8 kg Mt Everest 9000 m height,. Estimate volume as 1/3 h^3 = 1/3 * (9e3)^3 = 243 e 9 m^3. (if density = 1000 kg/m^3) = 2.43e14 kg. Yes the force will be the same if the product of the charges is the same Use x and y components

Example A solid insulating sphere of radius R has uniform volume charge density ρ and carries total charge Q. Find the flux through the outside sphere (r ≥ a). What if r < R? R x y Q r