Obtaining Electric Field from Electric Potential Assume, to start, that E has only an x component Similar statements would apply to the y and z components Equipotential surfaces must always be perpendicular to the electric field lines passing through them

2 For Three Dimensions In general, the electric potential is a function of all three dimensions Given V (x, y, z) you can find E x, E y and E z as partial derivatives

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6 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

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

8 V for a Continuous Charge Distribution, cont To find the total potential, you need to integrate to include the contributions from all the elements This value for V uses the reference of V = 0 when P is infinitely far away from the charge distributions

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12 V for a Uniformly Charged Sphere A solid sphere of radius R and total charge Q For r > R, For r < R,

13 V for a Uniformly Charged Sphere, Graph The curve for V D is for the potential inside the curve It is parabolic It joins smoothly with the curve for V B The curve for V B is for the potential outside the sphere It is a hyperbola

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V Due to a Charged Conductor Consider two points on the surface of the charged conductor as shown is always perpendicular to to the displacement Therefore, = 0 Therefore, the potential difference between A and B is also zero

20 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

21 E and V of a sphere conductor The electric potential is a function of r The electric field is a function of r 2 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

22 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

23 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

24 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

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

26 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

27 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 The SI unit of capacitance is a farad (F)

28 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

29 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

30 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

31 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

32 Capacitance – Parallel Plates, cont. The capacitance is proportional to the area of its plates and inversely proportional to the plate separation

33 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

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35 Capacitance – Isolated Sphere Assume a spherical charged conductor Assume V = 0 at infinity Note, this is independent of the charge and the potential difference

36 Capacitance of a Cylindrical Capacitor From Gauss’ Law, the field between the cylinders is E = 2 k e / r  V = -2 k e ln (b/a) The capacitance becomes

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

38 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

39 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 + Q 2 The potential difference across the capacitors is the same And each is equal to the voltage of the battery

40 Capacitors in Parallel, 3 The capacitors can be replaced with one capacitor with a capacitance of C eq The equivalent capacitor must have exactly the same external effect on the circuit as the original capacitors

41 Capacitors in Parallel, final C eq = C 1 + C 2 + … 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

42 Capacitors in Series When a battery is connected to the circuit, electrons are transferred from the left plate of C 1 to the right plate of C 2 through the battery

43 Capacitors in Series, 2 As this negative charge accumulates on the right plate of C 2, an equivalent amount of negative charge is removed from the left plate of C 2, 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

44 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

45 Capacitors in Series, final The equivalent capacitance of a series combination is always less than any individual capacitor in the combination

46 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

47 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

48 Equivalent Capacitance, Example The 1.0  F and 3.0  F are in parallel as are the 6.0  F and 2.0  F 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

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