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20.4 Obtaining Electric Field from Electric Potential Assume, to start, that E has only an x component 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 1 For Three Dimensions 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 2 3 4 5 Electric Field and Potential of 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 20.5 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 7 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 8 9 10 11 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 2 VD VC R r 2R 3 keQ r2 VD 3 2 3R R 12 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 13 14 15 16 17 18 20.6 V Due to a Charged Conductor Consider two points on the surface of the charged conductor as shown E is always perpendicular to to the displacement ds Therefore, E ds = 0 Therefore, the potential difference between A and B is also zero 19 V Due to a Charged Conductor, cont V is constant everywhere on the surface of a charged conductor in equilibrium DV = 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 20 E and V of a sphere conductor 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 21 Two charged sphere conductors connected by a conducting wire 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 22 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 is must be zero 23 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, VB VA E ds 0 A cavity surrounded by conducting walls is a field-free region as long as no charges are inside the cavity 24 20.7 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 25 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 26 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 DV The SI unit of capacitance is a farad (F) 27 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 (mF) and picofarads (pF) The capacitance of a device depends on the geometric arrangement of the conductors 28 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 29 Capacitance – Parallel Plates The charge density on the plates is s = 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 30 Parallel Plate Assumptions The assumption that the electric field is uniform is valid in the central region, but not at the ends of the 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 31 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 DV Ed Qd / o A d 32 A parallel-plate Capacitor connected to a Battery 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 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 33 34 Capacitance – Isolated Sphere Assume a spherical charged conductor Assume V = 0 at infinity Q Q R C 4 o R DV keQ / R ke Note, this is independent of the charge and the potential difference 35 Capacitance of a Cylindrical Capacitor From Gauss’ Law, the field between the cylinders is E = 2 ke l / r DV = -2 ke l ln (b/a) The capacitance becomes Q C DV 2ke ln b a 36 20.8 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 37 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 38 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 = Q 1 + Q2 The potential difference across the capacitors is the same And each is equal to the voltage of the battery 39 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 40 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 41 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 42 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 43 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 44 Capacitors in Series, final Q Q1 Q2 DV V1 V2 1 1 1 Ceq C1 C2 The equivalent capacitance of a series combination is always less than any individual capacitor in the combination 45 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 46 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 47 Equivalent Capacitance, Example The 1.0mF and 3.0mF are in parallel as are the 6.0mF and 2.0mF These parallel combinations are in series with the capacitors next to them The series combinations are in parallel and the final equivalent capacitance can be found 48 49 50