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Midterm 4 Statistics Bin count average median st dev hi lo 0 5 10 15 20 25 30 35 40 45 50 55 60 Frequency 0 1 1 7 15 12 30 44 52 61 40 26 4 60 pts 293 37 39 10.37904 58 3 100% 293 62.21133 65.00001 17.2984 95.83335 5.000001 70 60 50 40 30 20 10 0 0 5 Physics 2220 - Fall 2016 - Exam 4 Average=37 pts (62%) 10 15 20 25 30 35 40 45 50 55 60 FINAL REVIEW Review F1 A spherical non-conductor has a uniform volume charge density ρ and radius R. Given: R, ρ use Gauss’ law to find: (a) The electric field at a distance r > R (b) The electric field at a distance r < R (c) The potential difference between two points at distances y rB > R and rA < R. qin = rVin R x You need to split the integral into two parts Review F2 A uniformly charged rod of length L and charge Q has the shape of a semicircle. (a) Calculate the Electric field at the center of the circle P. (b) Find the work needed to bring a charge q from ∞ to P. dq E ke 2 rˆ r Only the y component is not zero. The x component is zero because of symmetry Q dq = l ds = ds L VP = ke ò dq r ds Rd V¥ = 0 DU = qDV Review F3 The switch S is closed for a long time and charges the capacitors. There is a dielectric κ in C3. (All capacitor values are given without dielectric). Find the charge and potential on each capacitor. S Q Cº V ε C C0 C1 C2 Qtotal = Ctotale Ctotal = (1/ C1 +1/ (C2 + k C3 ))-1 Q1 = Qtotal Q2 + Q3 = Qtotal V2 = V3 C3 κ Review F4 The switch is closed at t = 0. (a) Find the voltage across each resistor at t = 0. (b) Find the voltage across each resistor at t = ∞. (c) The capacitor is fully charged and the switch is opened. Calculate the time constant for discharging the capacitor. (d) Find the charge on the capacitor, Q(t), after the switch is opened. S R1 C ε R2 R3 Review F4 solution The switch is closed at t = 0. (a) At t = 0 the capacitor is like a wire. Redraw the circuit and find the currents. (b) At t = ∞ there is no current in R2. Redraw the circuit and find the currents. (c) Redraw the circuit without the battery and R1. S R1 t = (R2 + R3 )C (d) Find the charge on the capacitor, Q(t), after the switch is opened. Q(t) = Qmax e-t/t C ε R2 R3 Review F5 (a) Find the equivalent resistance of the circuit. R eq =15W (b) Find the current through the battery. I bat = I1 =15.0V / R eq (c) Find the potential difference across ac. Vac = R1I1 (d) Write Kirchhoff’s rules for the circuit. Review F6 A long coaxial cable has current +I on the interior cable and -I on the exterior cable. The current density J is uniform over the cross section of the interior conductor. (a) Calculate the magnetic field at r < a, a < r < b and r > b. (b) Calculate the magnetic energy stored between the cables in one meter of the coaxial cable. dI I J= = dA A uB =dU/dV=B2/2μ0 U = ò uB (r) dV dV = 2p rldr b U = 2p l ò uB (r)rdr a Review F7 Three very long straight wires carrying currents I1 = -10.0 A (into the paper) and I2 = 5.00 and I3 = 8.00 A are perpendicular to the paper. The side a is 10.0 cm. (a) Find the total magnetic field, magnitude and direction, at point P. (b) Find the total force per meter, magnitude and direction, on I3 exerted by the other two currents. m0 I B(r) = 2p r .P a Use right hand rule to find the directions and find BP,x I1 and BP,y components. a I2 m0 I1I 3 F13 = l 4p a m0 I 2 I3 F23 = l 2p a I3 Review F8 Two very long wires are parallel and a distance d apart. The wires carry a current I in opposite directions and its magnitude is given by I = I0 e-ωt. Given: I0, d and ω. (a) Find the magnetic field generated by the two wires as a function of time at a point P located between the wires, at a distance r from the left wire. (b) Calculate the magnetic flux through the rectangular wire loop shown as a function of time. (the dimensions l and a are given) (c) Calculate the induced emf in the loop as a function of time. d r -I .P a a l I Review F8 solution Two very long wires are parallel and a distance d apart. The wires carry a current I in opposite directions and its magnitude is given by I = I0 e-ωt. Given: I0, d and ω. d m0 I m0 I B(r) = + 2p r 2p r(d - r) r -I eind dFB =dt .P a a l I Review F9 A circular wire loop of radius r and resistance R turns about a vertical axis with constant angular speed ω in a region of constant uniform magnetic field B perpendicular to the rotation axis. At t = 0 the unit vector normal to the loop is in the direction of B. (a) Find the magnetic flux through the loop as a function of time. (b) Find the induced current in the loop as a function of time. (c) What is the net magnetic force on the loop? Review F9 solution A circular wire loop of radius r and resistance R turns about a vertical axis with constant angular speed ω in a region of constant uniform magnetic field B perpendicular to the rotation axis. At t = 0 the unit vector normal to the loop is in the direction of B. dFB e =dt I =e / R Review F10 The switch is connected to position a for a long time. At t = 0 the switch is thrown to position b. After this time find: (a) The frequency of the LC circuit. (b) The charge on the capacitor and the current and the voltage in the inductor as functions of time. w= 1 (c) The total energy of the circuit at t = 3.00 s. LC Qmax = eC Q(t) = Qmax cos(w t) I(t) = dQ dI , VL (t) = -L dt dt 1 Utot (t) = Ce 2 = constant 2