Coursehero >> Pennsylvania >> Drexel >> PHYS 102

Course Hero has millions of student submitted documents similar to the one below including study guides, homework solutions, papers, exam answer keys and textbook solutions.

reminder Quiz Quiz next Wednesday 8:00am 8:50 am Refer to course website for exam location No make-up quiz! 1 conceptual and 2 numerical problems Coverage: Chapter 19 Bring your calculator Basic formulas will be provided as well as universal physics constants Preparation: concentrate on L, R and HW problems as well as textbook examples; read the textbook Student help center for PHYS102 is open. 915 Disque Hall Hours: 10am-5pm M,Tue,W,Thu (check office hours schedule on the PHYS102 website 1 Problem 20.68. Four balls, each with mass m, are connected by four nonconducting strings to form a square with side a as shown in Figure P20.68. The assembly is placed on a horizontal, nonconducting, frictionless surface. Balls 1 and 2 each have charge q, and balls 3 and 4 are uncharged. Find the maximum speed of balls 3 and 4 after the string connecting balls 1 and 2 is cut. 2 Finding E From V Assume, to start, that E has only an x component r r dV (dV =) -E ds becomes E x dx and E x = - dx Similar statements would apply to the y and z components Equipotential surfaces must always be perpendicular to the electric field lines passing through them 3 E and V for an Infinite Sheet of Charge The equipotential lines are the dashed blue lines The electric field lines are the brown lines The equipotential lines are everywhere perpendicular to the field lines 4 E and V for a Point Charge The equipotential lines are the dashed blue lines The electric field lines are the brown lines The equipotential lines are everywhere perpendicular to the field lines 5 E and V for a Dipole The equipotential lines are the dashed blue lines The electric field lines are the brown lines The equipotential lines are everywhere perpendicular to the field lines 6 Electric Field from Potential, General In general, the electric potential is a function of all three dimensions Given V (x, y, z) you can find Ex, Ey and Ez as partial derivatives V Ex = - x V Ey = - y V Ez = - z 7 (i) In a certain region of space, the electric potential is zero everywhere along the x axis. From this information, we can conclude that the x component of the electric field in this region is 1. 2. 3. zero in the +x direction in the -x direction 33% 33% 33% ze in th e +x di re ct 10 8 io n ro In a certain region of space, the electric field is zero. From this information, we can conclude that the electric potential in this region is 1. 2. 3. 4. Zero Constant Positive Negative 25% 25% 25% 25% 10 ro e si tiv st a on Po C N eg at iv e Ze nt 9 Electric Potential for a Continuous Charge Distribution Consider a small charge element dq Treat it as a point charge The potential at some point due to this charge element is dq dV = ke r 10 V for a Continuous Charge Distribution, cont To find the total potential, you need to integrate to include the contributions from all the elements dq V = ke r This value for V uses the reference of V = 0 when P is infinitely far away from the charge distributions 11 V for a Uniformly Charged Ring P is located on the perpendicular central axis of the uniformly charged ring The ring has a radius a and a total charge Q dq = V = ke r k eQ 2 2 x +a 12 V for a Uniformly Charged Sphere A solid sphere of radius R and total charge Q Q For r > R, V = k e r For r < R, keQ 2 VD - VC = R -r2 2R 3 keQ r2 VD = 3 - 2 3R R ( ) 13 V for a Uniformly Charged Sphere, Graph The curve for VD is for the potential inside the curve It is parabolic It joins smoothly with the curve for VB The curve for VB is for the potential outside the sphere It is a hyperbola 14 V Due to a Charged Conductor Consider two points on the surface of the charged conductor as shown r E is always perpendicular to to the r displacement ds r r Therefore, E ds = 0 Therefore, the potential difference between A and B is also zero 15 V Due to a Charged Conductor, cont V is constant everywhere on the surface of a charged conductor in equilibrium V = 0 between any two points on the surface The surface of any charged conductor in electrostatic equilibrium is an equipotential surface Because the electric field is zero inside the conductor, we conclude that the electric potential is constant everywhere inside the conductor and equal to the value at the surface 16 E Compared to V The electric potential is a function of r The electric field is a function of r2 The effect of a charge on the space surrounding it The charge sets up a vector electric field which is related to the force The charge sets up a scalar potential which is related to the energy 17 Irregularly Shaped Objects The charge density is high where the radius of curvature is small And low where the radius of curvature is large The electric field is large near the convex points having small radii of curvature and reaches very high values at sharp points Example: lightning rod 18 Cavity in a Conductor Assume an irregularly shaped cavity is inside a conductor Assume no charges are inside the cavity The electric field inside the conductor must be zero 19 Cavity in a Conductor, cont The electric field inside does not depend on the charge distribution on the outside surface of the conductor For all paths between A and B, A cavity surrounded by conducting walls is a field-free region as long as no charges are inside the cavity 20 r r VB - VA = - E ds = 0 Capacitors Capacitors are devices that store electric charge The capacitor is the first example of a circuit element A circuit generally consists of a number of electrical components (called circuit elements) connected together by conducting wires forming one or more closed loops 21 Definition of Capacitance The capacitance, C, of a capacitor is defined as the ratio of the magnitude of the charge on either conductor to the potential difference between the conductors Q C V The SI unit of capacitance is a farad (F) 22 Makeup of a Capacitor A capacitor consists of two conductors When the conductors are charged, they carry charges of equal magnitude and opposite directions A potential difference exists between the conductors due to the charge The capacitor stores charge 23 More About Capacitance Capacitance will always be a positive quantity The capacitance of a given capacitor is constant The capacitance is a measure of the capacitor's ability to store charge The Farad is a large unit, typically you will see microfarads (F) and picofarads (pF) The capacitance of a device depends on the geometric arrangement of the conductors 24 Problem 20.32. Two conductors having net charges of +10.0 C and 10.0 C have a potential difference of 10.0 V between them. (a) Determine the capacitance of the system. (b) What is the potential difference between the two conductors if the charges on each are increased to +100 C and 100 C? 25 Capacitance Isolated Sphere Assume a spherical charged conductor Assume V = 0 at infinity Q Q R C= = = = 4 o R V keQ / R ke Note, this is independent of the charge and the potential difference 26 Parallel Plate Capacitor Each plate is connected to a terminal of the battery If the capacitor is initially uncharged, the battery establishes an electric field in the connecting wires 27 Capacitance Parallel Plates The charge density on the plates is = Q/A A is the area of each plate, which are equal Q is the charge on each plate, equal with opposite signs The electric field is uniform between the plates and zero elsewhere 28 Capacitance Parallel Plates, cont. The capacitance is proportional to the area of its plates and inversely proportional to the plate separation o A Q Q Q = = = C= V Ed Qd / o A d 29 Parallel Plate Assumptions The assumption that the electric field is uniform is valid in the central region, but not at the ends the of plates If the separation between the plates is small compared with the length of the plates, the effect of the non-uniform field can be ignored 30 Energy in a Capacitor Overview Consider the circuit to be a system Before the switch is closed, the energy is stored as chemical energy in the battery When the switch is closed, the energy is transformed from chemical to electric potential energy 31 Energy in a Capacitor Overview, cont The electric potential energy is related to the separation of the positive and negative charges on the plates A capacitor can be described as a device that stores energy as well as charge 32 Capacitance of a Cylindrical Capacitor From Gauss' Law, the field between the cylinders is E = 2 ke / r V = -2 ke ln (b/a) The capacitance becomes l Q C= = V 2ke ln b ( a ) 33 Circuit Symbols A circuit diagram is a simplified representation of an actual circuit Circuit symbols are used to represent the various elements Lines are used to represent wires The battery's positive terminal is indicated by the longer line 34 Capacitors in Parallel When capacitors are first connected in the circuit, electrons are transferred from the left plates through the battery to the right plate, leaving the left plate positively charged and the right plate negatively charged 35 Capacitors in Parallel, 2 The flow of charges ceases when the voltage across the capacitors equals that of the battery The capacitors reach their maximum charge when the flow of charge ceases The total charge is equal to the sum of the charges on the capacitors Q = Q1 + Q2 The potential difference across the capacitors is the same And each is equal to the voltage of the battery 36 Capacitors in Parallel, 3 The capacitors can be replaced with one capacitor with a capacitance of Ceq The equivalent capacitor must have exactly the same external effect on the circuit as the original capacitors 37 Capacitors in Parallel, final Ceq = C1 + C2 + ... The equivalent capacitance of a parallel combination of capacitors is the algebraic sum of the individual capacitances and is larger than any of the individual capacitances 38 Problem 20.39. Two capacitors, C1 = 5.00 F and C2 = 12.0 F, are connected in parallel, and the resulting combination is connected to a 9.00V battery. (a) What is the equivalent capacitance of the combination? What are: (b) the potential difference across each capacitor and (c) the charge stored on each capacitor? 39 Capacitors in Series When a battery is connected to the circuit, electrons are transferred from the left plate of C1 to the right plate of C2 through the battery 40 Capacitors in Series, 2 As this negative charge accumulates on the right plate of C2, an equivalent amount of negative charge is removed from the left plate of C2, leaving it with an excess positive charge All of the right plates gain charges of Q and all the left plates have charges of +Q 41 Capacitors in Series, 3 An equivalent capacitor can be found that performs the same function as the series combination The potential differences add up to the battery voltage 42 Capacitors in Series, final Q = Q1 + Q2 + K V = V1 + V2 + K 1 1 1 = + +K Ceq C1 C2 The equivalent capacitance of a series combination is always less than any individual capacitor in the combination 43 Summary and Hints Be careful with the choice of units In SI, capacitance is in F, distance is in m and the potential differences in V Electric fields can be in V/m or N/c When two or more capacitors are connected in parallel, the potential differences across them are the same The charge on each capacitor is proportional to its capacitance The capacitors add directly to give the equivalent capacitance 44 Summary and Hints, cont When two or more capacitors are connected in series, they carry the same charge, but the potential differences across them are not the same The capacitances add as reciprocals and the equivalent capacitance is always less than the smallest individual capacitor 45 Problem 20.41. Four capacitors are connected as shown in Figure P20.41. (a) Find the equivalent capacitance between points a and b. (b) Calculate the charge on each capacitor, taking Vab = 15.0 V. 46 Energy Stored in a Capacitor Assume the capacitor is being charged and, at some point, has a charge q on it The work needed to transfer a charge from one plate to the other is The total work required is W = Q 0 q dW = Vdq = dq C q Q2 dq = 2C C 47 Energy, cont The work done in charging the capacitor appears as electric potential energy U Q2 1 1 U= = QV = C( V )2 2C 2 2 This applies to a capacitor of any geometry The energy stored increases as the charge increases and as the potential difference increases In practice, there is a maximum voltage before discharge occurs between the plates 48 Energy, final The energy can be considered to be stored in the electric field For a parallel plate capacitor, the energy can be expressed in terms of the field as U = (oAd)E2 It can also be expressed in terms of the energy density (energy per unit volume) uE = o E2 49 Capacitors with Dielectrics A dielectric is an insulating material that, when placed between the plates of a capacitor, increases the capacitance Dielectrics include rubber, plastic, or waxed paper With a dielectric, C = Co The capacitance is multiplied by the factor when the dielectric completely fills the region between the plates For a parallel plate capacitor, this becomes C = = o(A/d) 50 Dielectrics, cont In theory, d could be made very small to create a very large capacitance In practice, there is a limit to d d is limited by the electric discharge that could occur though the dielectric medium separating the plates For a given d, the maximum voltage that can be applied to a capacitor without causing a discharge depends on the dielectric strength of the material 51 Dielectrics, final Dielectrics provide the following advantages Increase in capacitance Increase the maximum operating voltage Possible mechanical support between the plates This allows the plates to be close together without touching This decreases d and increases C 52 Dielectrics An Atomic View The molecules that make up the dielectric are modeled as dipoles The molecules are randomly oriented in the absence of an electric field 53 Dielectrics An Atomic View, cont An external electric field is applied This produces a torque on the molecules The molecules partially align with the electric field 54 Dielectrics An Atomic View, final An external field can polarize the dielectric whether the molecules are polar or nonpolar The charged edges of the dielectric act as a second pair of plates producing an induced electric field in the direction opposite the original electric field 55 Table of Some Dielectric Values 56 Types of Capacitors Tubular Metallic foil may be interlaced with thin sheets of paper or Mylar The layers are rolled into a cylinder to form a small package for the capacitor 57 Types of Capacitors Oil Filled Common for high voltage capacitors A number of interwoven metallic plates are immersed in silicon oil 58 Types of Capacitors Variable Variable capacitors consist of two interwoven sets of metallic plates One plate is fixed and the other is moveable The capacitor generally vary between 10 and 500 pF 59 Types of Capacitors Electrolytic Is used to store large amounts of charge at relatively low voltages The electrolyte is a solution that conducts electricity by virtue of motion of ions contained in the solution 60 Problem 20.54. (a) How much charge can be placed on a capacitor with air between the plates before it breaks down if the area of each of the plates is 5.00 cm2? (b) Find the maximum charge assuming polystyrene is used between the plates instead of air. 61

Find millions of documents here - Study Guides, Homework Solutions, Papers, Exam Answer Keys and more. Course Hero has millions of course related materials that will enable you to learn better, faster and get an A in all your courses.
Below is a small sample set of documents: