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Transcript

Physics C Electric Potential Is the electrical force conservative or nonconservative? Name:__________________ Electric Potential Definition: What characterizes a conservative force? Mathematical relationship to potential energy: How does the change in a electrical potential energy relate to work? Write an equation that expresses the relationship between electrical potential energy and work done by an electrostatic force. High Potential (voltage) + + + + + A + + + + + + + + + + + Field Refresher Equi-potential Surfaces B + C + Sample Problem: A proton is accelerated through a potential difference of -20,000 V. What is the potential and kinetic energy change of the proton? How much work did the electric field do on the proton? Low Potential (voltage) - Sample Problem: How fast is a 2 MeV alpha particle moving? What potential difference is needed to stop this alpha particle? Electric Field The work done by an electrical force in moving a charge q from point A to point B is given by: Uniform field: Sample Problem: An electric field is given by E = 250 i V/m. A 12.0 C charge moves from the origin to (0.20 m, 0.50 m). What is the change in potential energy and electric potential for the charge? Non-uniform field: This results in a potential energy change of Uniform field: Non-uniform field: 5/7/2017 1 Bertrand Sample Problem: An electron is released from rest in an electric field of magnitude by 5900 V/m. Through what potential difference will it have passed after moving 1.00 cm? How fast will it be moving at this time? Sample Problem: Two point charges (5.0 nC and -3.0 nC) are separated by 35 cm. What is the potential energy of the pair? What is the electric potential at a point midway between the charges? Equi-potential or Iso-potential surfaces Definition: Mathematically, how do you determine the electric field from the electric potential if you know how the potential changes with position? Characteristics: Draw the field lines and equi-potential surfaces around a positive point charge What can you say about how field strength relates to the rate at which the electric potential is changing? Sample Problem: Find the potential at a distance of 1.00 cm from a proton. Repeat for an electron. 5/7/2017 Sample Problem: The electric potential in a region is given by the function V = -9.0 x V/m – 3.0 x2 V/m2. What is the magnitude and direction of the electric field at x = 2.0 m? 2 Bertrand Sample Problem: Over a region of space, the electric potential is given by V = -3.0 x + 6.0 x y – 2.0 x2y. Derive the vector representing the electric field at (1, 2) m. Sample Problem: Find the potential at a point inside a charged non-conducting solid sphere of radius R as a function of its distance from the center of the sphere. Assume charge Q is distributed uniformly. Equation for potential due to multiple point charges: Where do conductors bear excess charge? Equation for potential due to multiple point charges: Does it matter if the conductor is solid or hollow? Sample Problem: Determine the electric potential at a point P located on the perpendicular axis of a uniformly charged ring of radius R and total charge Q. Sample Problem Two charged conductors are connected by a long conducting wire, and a charge of 10 nC is placed on the combination. Sphere A has a diameter of 10 mm, and sphere B has a diameter of 5 mm. How much charge is on each sphere? What is the electric potential of each sphere? Sample Problem: Determine the electric potential along the perpendicular central axis of a uniformly charged disk of radius R and surface charge density . 5/7/2017 3 Bertrand What is a Capacitor? Parallel plate capacitance equation: Deriving Capacitance -- Steps What does it mean when we say a capacitor is “charged”? 1. 2. 3. 4. 5. 6. Draw a parallel plate capacitor Draw the capacitor; identify symmetry Draw Gaussian surface Write Gauss’ Law Solve Gauss’ Law for E Develop function for V from E Develop function for C from V Derive C for Parallel Plate Capacitor Draw a cylindrical capacitor Derive C for Charged Sphere Capacitance equation: Upon what does capacitance depend? Sample Problem: How much charge is on each plate of a 4.00 connected to a 12-V battery? 5/7/2017 4 Bertrand Equivalent capacitance – series capacitors Derive C for a Spherical Capacitor What is the same for all capacitors in series? Why is the above statement true? Equivalent capacitance – parallel capacitors What is the same for all capacitors in parallel? Why is the above statement true? Derive C for a Cylindrical Capacitor Sample problem: Determine equivalent capacitance of the configuration shown C C C 5/7/2017 5 C C C Bertrand What is a dielectric? Sample problem: Determine equivalent capacitance between a and b. If the potential difference between a and b is 10V, what charge is stored on C3? (C1=5F, C2=10F, C3=2F) C1 a C1 C3 C2 C2 C2 How is capacitance changed when a dielectric is inserted between the capacitor plates? Write the relationship here. C2 b How does a dielectric work? How do capacitors store energy? Sample Problem Find the capacitance of a parallel plate capacitor that uses Bakelite as a dielectric, if each of the plates has an area of 5.0 cm 2 and the plate separation is 2.00 mm. Equation for energy stored in capacitor: Sample Problem: A 3.00 mF capacitor is connected to a 12.0 V battery. How much energy is stored in the capacitor? 5/7/2017 6 Bertrand Physics C Electric Potential Name:__________________ FR Problem 1 A006 E1. The square of side a at right contains a positive point charge +Q fixed at the lower left corner and negative point charges -Q fixed at the other three corners of the square. Point P is located at the center of the square. a. On the diagram, indicate with an arrow the direction of the net electric field at point P. b. Derive expressions for each of the following in terms of the given quantities and fundamental constants. i. The magnitude of the electric field at point P ii. The electric potential at point P c. A positive charge is placed at point P. It is then moved from point P to point R, which is at the midpoint of the bottom side of the square. As the charge is moved, is the work done on it by the electric field positive, negative, or zero? Positive Negative Zero Explain your reasoning. d. i: Describe one way to replace a single charge in this configuration that would make the electric field at the center of the square equal to zero. Justify your answer. ii. Describe one way to replace a single charge in this configuration such that the electric potential at the center of the square is zero but the electric field is not zero. Justify your answer. 5/7/2017 7 Bertrand FR Problem 2 S116 E1. Consider the electric field diagram above. a. Points A, B, and C are all located at y = 0.06 m . i. At which of these three points is the magnitude of the electric field the greatest? Justify your answer. ii. At which of these three points is the electric potential the greatest? Justify your answer. b. An electron is released from rest at point B. i. Qualitatively describe the electron's motion in terms of direction, speed, and acceleration. ii. Calculate the electron's speed after it has moved through a potential difference of 10 V. c. Points B and C are separated by a potential difference of 20 V. Estimate the magnitude of the electric field midway between them and state any assumptions that you make. d. On the diagram, draw an equipotential line that passes through point D and intersects at least three electric field lines. 5/7/2017 8 Bertrand FR Problem 3 S336E1. A spherical cloud of charge of radius R contains a total charge +Q with a non-uniform volume charge density that varies according to the equation (r) = o(1 – r/R) for r < R and = 0 for r > R, where r is the distance from the center of the cloud. Express all algebraic answers in terms of Q, R, and fundamental constants. a. Determine the following as a function of r for r > R . i. The magnitude E of the electric field ii. The electric potential V b. A proton is placed at point P shown above and released. Describe its motion for a long time after its release. c. An electron of charge magnitude e is now placed at point P, which is a distance r from the center of the sphere, and released. Determine the kinetic energy of the electron as a function of r as it strikes the cloud. d. Derive an expression for o . e. Determine the magnitude E of the electric field as a function of r for r < R . 5/7/2017 9 Bertrand FR Problem 4 S118E2. In the figure at right, a non-conducting solid sphere of radius a with charge +Q uniformly distributed throughout its volume is concentric with a non-conducting spherical shell of inner radius 2a and outer radius 3a that has a charge –Q uniformly distributed throughout its volume. Express all answers in terms of the given quantities and fundamental constants. (a) Using Gauss's law, derive expressions for the magnitude of the electric field as a function of radius r in the following regions. i. Within the solid sphere (r < a ) ii. Between the solid sphere and the spherical shell (a < r < 2a ) iii. Within the spherical shell (2a < r < 3a ) iv. Outside the spherical shell (r > 3a ) (b) What is the electric potential at the outer surface of the spherical shell (r = 3a )? Explain your reasoning. (c) Derive an expression for the electric potential difference Vx – Vy between points X and Y shown in the figure. 5/7/2017 10 Bertrand FR Problem 5 A277E1. An isolated conducting sphere of radius a = 0.20 m is at a potential of -2,000 V. a. Determine the charge Q0 on the sphere. The charged sphere is then concentrically surrounded by two uncharged conducting hemispheres of inner radius b = 0.40 m and outer radius c = 0.50 m, which are joined together as shown above, forming a spherical capacitor. A wire is connected from the outer sphere to ground, and then removed. b. Determine the magnitude of the electric field in the following regions as a function of the distance r from the center of the inner sphere. i. r <a ii. a < r < b iii. b < r < c iv. r > c c. Determine the magnitude of the potential difference between the sphere and the conducting shell. d. Determine the capacitance of the spherical capacitor. 5/7/2017 11 Bertrand FR Problem 6 A277E3. The nonconducting ring of radius R shown at right lies in the yz-plane and carries a uniformly distributed positive charge Q. a. Determine the electric potential at points along the x-axis as a function of x. b. i. Show that the x-component of the electric field along the x-axis is given by Ex Qx 3 4 0 R 2 x 2 2 ii. What are the y- and z- components of the electric field along the x-axis? c. Determine the following. i. The value of x for which Ex is a maximum ii. The maximum electric field Ex max d. On the axes below, sketch Ex versus x for points on the x-axis from x = -2R to x = +2R. e. An electron is placed at x = R/2 and released from rest. Qualitatively describe its subsequent motion. 5/7/2017 12 Bertrand Use this page for Extra Notes 5/7/2017 13 Bertrand Use this page for Extra Notes 5/7/2017 14 Bertrand