Electric Potential. Capacitors (Chapters 28, 29)

Electric Potential. Capacitors (Chapters 28, 29) Electric potential energy, U Electric potential energy in a constant field Conservation of energy Electric potential, V Relation to the electric field strength The potential of a point charge Calculating the potential of Multiple point charges Continuous distributions of charge Capacitors Capacitance, C Simple circuits Energy Dielectrics

Electric Potential Energy Constant electric field The electrostatic force is a conservative force, so it is associated with an electrical potential energy, U: decreasing from positive to negative charges To find a qualitative expression, recall that the work done by a conservative force is always equal to the negative of the change in potential energy For simplicity, let s start with the particular case of a uniform electric field E like between two metallic parallel plates charged with unlike charges (that is, a capacitor) Consider a positive charge q moving parallel with the field E from a to b: the field does positive work on it W F x qex Therefore, the change in electrical potential energy when the field is uniform is U U U W qex b e a How is this useful in describing the motion of the charged particle? q a Fe E qe Δx b

Electric Potential Energy Spontaneous Charge Movements In nature, systems allowed to evolve without constraints will have the tendency to adopt configurations corresponding to a minimum of energy Accordingly, if a charge is allowed to move spontaneously in an electric field, the motion will be such that the electric potential energy decreases, that is Therefore, if the field E is uniform and positive (that is, it points to the right, in positive x direction), we have U qex q q U U U 0 x 0 0 x 0 f Positive charge moves to the right (same as the direction of the field) Notice that this behavior derived by analyzing the spontaneous change in potential energy is consistent with the direction of motion expected based on the electric force acted on the charge by the electric field i 0 E F F Negative charge moves to the right (against the direction of the field) x x

Electric Potential Energy Energy conservation In conclusion, any electric charge released in the electric field experiences a force and accelerates, gaining kinetic energy K on behalf of the electric potential energy of the charge-field combination Conservation of net energy still works as discusses in PHY181, except that the potential energy U must also include the electric potential energy (besides the gravitational and elastic potential energies): net potential energy including gravitational, elastic and electric K U W nonconservative Hence, if there is no nonconservative force (like friction), we have K U = const. Ex: The motion of a charge in a constant electric field can be compared to the motion of a mass in the constant gravitational field: A positive charge q moving a vertical distance Δy loses electric potential energy qeδy, which gets converted into kinetic energy A mass m falling a vertical distance Δy loses gravitational potential energy mgδy, which gets converted into kinetic energy Quiz 1: A positive charge of 1 C moves 1 m in a uniform field E. What is the field E if the charge gains the same kinetic energy as a 1 kg mass freely falling 1 m? a) 1.0 N/C b) 9.8 N/C c) 4.9 N/C Δy electric q Fe qe K qey gravitational m Fg mg K mgy

Electric Potential Potential difference In order to describe the energy of the electric field rather than the energy of a particular charge in the field, we introduce Def: The electric potential V is the electric potential energy per unit charge Hence, the potential difference ΔV between points a and b in an electric field is the change in the potential energy of a test charge q moved from a to b divided by the charge: U V V Va Vb U qv Volts V J C SI q Both electric potential energy and potential difference are scalar quantities Notice that, by multiplying the charge with the potential difference it moves through, we obtain the energy exchanged between the charge and the field in the process Ex: In a capacitor, the field E is uniform, so the potential difference between any two points a-b at a separation Δx along the field is given by V Ex q ab Va Vb U q q Ex a b Quiz 2: What is the potential difference between the positive plate and the point c inside the capacitor represented above? a) E/2d b) Ed c) E2d d) Ed/2 V V ΔV ab Δx E c d

Electric Potential Comments Since the potential is not absolute, in general we will be working with potential differences: Later we ll even denote it V, and we ll call it voltage, but we ll still refer to a potential difference Then, if a charge moves through a difference of potential between two points a and b, the work done on it by the electric field is W qv q V V b Ex: A 1.5 V battery has a potential difference between its terminals of 1.5V, such that the work done by the battery to move a unit positive charge across it is 1.5 J. a U V V V V q A positive charge released in a field will move from a high potential to a low potential, while a negative charge will move from low to high b a ab alternative notation High V W qv 0 V 0 E W qv 0 V 0 Low V (ground)

Electric Potential Electric field and electric potential The electric potential is the potential energy per unit charge in the same way the electric field is the electric force per unit charge: so, while the force and the potential energy depends on the test charge q 0, the field and the potential depend only on the source of field: Hence, everywhere in the space surrounding a charge, each point is characterized by a field E (vector) and a potential V (scalar). (However, the field strength is absolute while the potential is relative to a ground where the potential is set to be zero) Based on the definition ΔU = qδv = W, we see that, if a particle moves between two points in space with the same potential, the electric force will do zero work In conclusion: E F q 0 V U q 0 1. When a positive charge is placed in an electric field It moves in the direction of the field The electrical potential energy decreases It moves from a point of higher potential to a point of lower potential (ΔV < 0) Its kinetic energy increases U qv 2. When a negative charge is placed in an electric field It moves opposite to the field The electrical potential energy decreases It moves from a point of lower potential to a point of higher potential (ΔV > 0) Its kinetic energy increases U qv

Problem: 1. Motion of a proton in an electric field: An proton moves 2.5 cm parallel to a uniform electric field of E = 200 N/C. Assume the electric field in the positive direction. a) How much work is done by the field on the proton? b) What change occurs in the potential energy of the proton? c) What potential difference did the proton move through? d) If the proton is released from rest what is its final speed?

Electric Potential Relationship between E and V Since everywhere in the space surrounding a charge, each point is characterized by a field E (vector) and a potential V (scalar), it is natural to ask: Is there a relationship between the two? Yes, as following: 1. Consider a positive test charge q in an electric field: since the vector electric field is tangent to the line, the electric force on q is also tangent 2. Say that q is moved a small step dx a) perpendicular on the field line, the field won t do any work: du = 0 dv = 0 b) parallel with the field line, the field will do a maximum work: du = max dv = max dv 0 dv 0 dv 0 du dw Fe dx cos qedx cos 0, q Edx cos qdv dv max Edx du qdv 4. Hence, if a field is given by its position dependent potential, dv and V(x) is plotted along a certain x-axis, the field in every E point is given by the negative of the slope of the V(x) graph dx dx dv max E F qe electric field line 3. Therefore, the vector field is oriented such that the change in potential is maximum. Demonstration: for an elementary step dx in the field making an angle θ with the force dv max θ q slope

Electric Potential Potential gradient The fact that the electric field points in the direction corresponding to the fastest decrement in potential can be based on the observation we made in PHY181 about how any conservative force is given by the gradient of its potential energy In the case of electric forces we have: du du du dv dv dv Fe,, qe q,, dx dy dz dx dy dz such that E dv dx, E dv dy, E dv dz x y z E V gradient Conversely, when the field is integrated between two points, one obtains the potential difference between the two points, if we know the potential, we can calculate the field and vice-versa dv Ex: For a radial field Er dv Erdr dr Quiz 3: How does the potential depend on the position in the interior of any statically charged conductor? a) It is always zero b) It increases from center to the surface of the conductor c) It is constant

Electric Potential Charged Conductors In the previous chapter, we learned that the electric field inside a conductor in electrostatic equilibrium is zero. What about the electric potential? All points on the surface of a charged conductor in electrostatic equilibrium are at the same potential which can be taken by convention to be zero (ground) A volume, or surface, or line with points at the same electric potential is called as having an equipotential As shown by the relationship between E and V, the electric field at every point on an equipotential surface is perpendicular to the surface, and the lines of electric field are everywhere perpendicular on equipotential lines Ex: Consider an arbitrarily shaped conductor with an excess of positive charge in electrostatic equilibrium All of the charge resides at the surface, predominantly on pointier sides The electric field is zero inside the conductor, and nonzero and perpendicular on the surface just outside the surface The electric potential is a constant everywhere on the surface of the conductor, so no work is necessary to move charges on the surface between any two points A and B: consistent with the fact that the field and so the electric force on the moving charge is perpendicular on the surface The potential everywhere inside the conductor is constant and equal to its value at the surface so the bulk is an equipotential volume

Electric Potential Potential energy of a pair of point charges The work done by a field created by a point charge q on a charge q 0 moved between two points is, for any path l (since electric forces are conservative), 2 2 W F d F d cos F dr kqq e e e r Therefore, the potential energy of the point charge q 0 at distance r from q is r 1 1 1 1 W kqq U U 0 2 1 r2 r1 r dr 0 2 r r 0 k U r Recall that the potential energy doesn t make sense if it not with respect to a certain point, a ground where the potential energy is zero So, since the expression above is zero when r : Fe qq r dr θ d q 0 r 1 E path q r 2 V 1 V 2 Def: The electric potential energy of a pair of point charges separated by a distance r is equal to the work done by one of the charges to bring the other one from infinity to the distance r.

V r Comments: U r q 0 Electric Potential Point Source Then, based on the definition of the potential, we can find the potential produced by a point charge source q at a distance r, taking the ground to be at an infinite distance from the source charge: k q r 1 q 4 r This expression gives us the work that the field would do in order to bring a unit charge from infinitely far away to a distance r from q Since it is associated with the electric field, a potential exists at some point in space irrespective if there is a test charge at that point Unlike the electric field which decreases like 1/r 2, the electric potential decreases like 1/r A positive charge creates a positive potential and a negative charge a negative potential V relative to a point at r = 0 q V 1 E k q r 2 q k r r 1 1 q E Ex: The field strength E in the vicinity of a positive charge at position r = 0 decreases faster than the respective potential V V r 2 V q k r V 1 V 2 slope of the V(r) curve for every r 2 q k r 2

Equipotential Maps Another way to visually represent fields The equipotential surfaces and lines separated by equal potential steps can be used to complement the electric field lines to picture out the electric field. For instance: 1. Point charges: the equipotential surfaces are a family of spheres centered at the point charge: the potential decreases in magnitude on larger and larger spheres, and the electric field lines are everywhere perpendicular to the equipotential surfaces 90 The field is oriented in the direction of maximum V change equipotential spheres V increasing (peak) V decreasing (sink) The stronger is the field, the faster the potential varies, and the closer are the equipotential lines Notice that the equipotential lines can be compared with the elevation contours on a topographic map: thus, the potential around a positive charge forms a peak of potential and around a negative charge a well (or sink) of potential

Equipotential Maps Two Point Charges 2. Dipole: if the map of potential is seen from above, the contours of equipotential lines appear projected on a plane (x-y) If a third axis perpendicular on the plane of the dipole is considered representing the potential V, the map has potential peaks in the position of positive charges and wells in the position of negative charges Notice that a profile projection of the map shows the potential variation with position in the V-x plane V y Top view V x x y x Side view

Quiz 4: An electron is released from rest at x = 2 m in the potential shown. What does the electron do right after being released? a) Stays at x = 2 m. b) Move to the right (x) at steady speed. c) Move to the right with increasing speed. d) Move to the left (x) at steady speed. e) Move to the left with increasing speed. Quiz 5: Which of the sets of equipotential surfaces below a horizontal electric field given on the right? (c is a constant) a) b) c) E cxiˆ d) e) f)

Electric Potential Multiple point charges and continuous sources 1. Multiple point charges: we can apply once again the Superposition Principle: The total electric potential at some point P due to n point charges at distances R i from P is the algebraic sum of the electric potentials due to the individual charges V net 1 q n i Vi i 4 0 i1 Ri The algebraic sum is used because potentials are scalar quantities simpler calculations than in the case of net electric fields. 2. Continuous distribution of charges: we ll consider only sources with very simple shapes and uniform distribution of charges. Then, the potential in a point of coordinates (x,y,z) is: V 1 dq 1 dq 4 0 R 4 0 x x y y z z 2 2 2 where R is the distance to each element of charge at position (x,y,z ). Notice that the integration is simpler than in the case of the field strength since there is no vector involved.

Problems: 2. Net electric potential by superposition: Three charges with magnitudes and signs given on the figure in terms of q = 3 μc are located in three corners of a square of side a = 2 cm. a) Find the net electric potentials V 1,2 in the center of the square, and at the upper right corner of a square b) A test charge q/2 is moved between the center of the square and the upper right corner. Calculate the work done by the net field of the other charges on the test charge. 3q a a V 2 =? V 1 =? q 2q 3. Continuous linear charge - ring: Consider once again the uniformly charged ring in the xy-plane. The ring has radius a and a charge q distributed evenly along its circumference. a) Calculate the potential in point P (0,0,h) above its center. b) Comment on what happens if the distance h is much larger than the radius a. 4. Continuous linear charge - line: Consider a charge q distributed with uniform density along y-axis, in the interval (a, a). a) Write out the charge dq per element of length dy of the line in terms of q and a. b) Calculate the potential in a point P (r,0) in terms of k, q, and radial distance r. Useful integral: a a 2 2 12 r a a dy ln 2 2 1 2 2 2 1 2 r y r a a

Problems: 5. Potential of a point charge: Knowing the electric field of a point charge in a point r, reobtain the potential at that position. 6. Potential inside a capacitor revisited: Knowing the electric field, calculate the potential in between the plates of a parallel plate capacitor with a surface charge density σ. 7. Metallic cylinder: Consider a charge deposited on a very long long metallic cylinder with linear density λ. a) Use the electric field calculated using Gauss s law to find the difference of the potential between two points outside the cylinder at distances r 1 and r 2 from its axis. b) Where would be a suitable ground for this particular potential? 8. Metallic sphere: Consider a charge q deposited on a metallic sphere see figure. Confirm the adjacent graphical representations by calculating the potential inside and outside the sphere using the electric field calculated using Gauss s Law?

Capacitors What are they? A capacitor is an electric device used in a variety of electric circuits Its functionality is based on the storage of energy associated with the electric field between two symmetric distributions of unlike charges insulated from one another Any two closely separated conductors will form a capacitor: in order to charge it, one can use a battery to do work in order to transfer a charge Q from one conductor to the other, such that one conductor will have a deficit Q and the other a surplus Q of electrons. As a result an electric field will appear between the conductors: electric field means ability to do work, or stored energy Capacitors come in various arrangements of conductors: parallel plates, concentric spheres, coaxial cylinders etc While developing the generic ideas about any such capacitor architectures, in PHY 182 we ll focus on the simplest (and most common) type: the parallel plate capacitor

Capacitors Capacitance The ability of a capacitor to store charge (that is electric field and energy) is given by its capacitance: Def: The capacitance, C, of a capacitor is defined as the ratio between the amount of electric charge Q it holds and the potential difference V between its plates C Q V C 1 Farad (F) 1C 1V SI Comments: One Farad is a very large capacitance: so most often we ll see µf, nf or pf The capacitance of a capacitor is a characteristic of the device, and does not depend on the difference of potential applied across the plates Thus, according to the definition, if a difference of potential V is applied across the plates of a capacitor of capacitance C, it will store a maximum charge Q = CV Q V Q Q CV C V V CEd E d

Problem: 9. Spherical capacitor: A spherical capacitor consists in an interior sphere of radius r a in the center of a spherical shell of inner radius r b. Calculate the capacitance in terms of r a, r b and constants.

For a parallel-plate capacitor filled with air, we can easily derive the capacitance by applying the definition to a capacitor as on the adjacent figure If under a potential difference V ab = Ed, the plates will store a charge of density σ, such that Q = σa. Then ab Capacitors The parallel-plate capacitor So, the capacitance depends on the geometric arrangement of the conductors and the electric properties of the insulating material between them. But how? C Q Q A V Ed d Ex: Consider a parallel-plate capacitor of area A, of plate separation d When connected to the battery of voltage V across battery, charge is pulled off one plate and transferred to the other plate The transfer stops when across capacitor across battery Then the charge stored on the capacitor plates will be V A Q CV V d 0 C V 0 across capacitor 0 A d

Capacitors Revisit the field between the parallel plates If the plate separation is much smaller than the size of the plates, the electric field inside is well approximated by the field of two infinite parallel sheets of charge However, the approximation works only close to the center of the plates, not near the edges where the field is not uniform Since the potential difference across the plates is V = Ed, if we apply a larger voltage V, a larger field is produces corresponding to more charge Q deposited Since the field is constant, the equipotential surfaces are equally spaced flat surfaces parallel with the plates The equipotential surface close to the positive plate has the largest potential, and the potential decreases uniformly to the surface on the negative plate: Ex: The potential differences V i V i 1 between equipotential surfaces at 4 equal steps Δx between the plates of a capacitor are EΔx, such that potential difference across the plates is V V V V V V V V V V V Ex Ed ab a b a 1 1 2 2 3 3 b 4 Potential V a V 1 V 2 V 3 V b V a > V 1 > V 2 > V 3 > V b slope = ΔV/Δx = E E Δx d x

Electric Circuits Capacitors in circuits A circuit is a network of electric devices usually containing a source of electrical energy (such as a battery) connected to electric elements (such as capacitors) A circuit diagram can be used to show the path of the real circuit If a capacitor is connected in a circuit across two points with an electric potential difference, the electrons are transferred through wires from one plate to the other plate, leaving one plate positively charged and the other plate negatively charged The flow of charges ceases when the voltage across the capacitor equals that across the two points in the circuits. Then, as long the potential difference remains unchanged, the capacitor stays inactive: a storage of charge (that is, electric energy) The capacitors are represented in circuits using a symbol for the two plates. The battery needed to produce potential differences is represented by a similar symbol The similarity is due to the fact that both devices are sources of electric charge; however, while the battery is ideally a limitless source, the capacitor is limited by its capacity C _ Capacitors can be combined in circuits: the simplest combinations are in parallel and series. Let s find the equivalent capacitance that performs the same function as these elementary combinations V

Electric Circuits Capacitors in parallel Capacitors in parallel are all connected across the same two points. For illustration, consider two capacitor in parallel Therefore, all capacitors will be connected across the battery, so they will be under the same voltage V The total charge is equal to the sum of the charges on the capacitors Q Q Q net 1 2 This net charge can be considered as being stored on the parallel combination seen as only one capacitor with an equivalent capacitance C p : Q Q Q net 1 2 C C C Cp V CV 1 CV 2 p 1 2 The result can be extrapolated to for n capacitors in parallel: C C C C C p 1 2 3... Notice that the parallel equivalent capacitance is larger than any of the individual capacitances n

Electric Circuits Capacitors in series Capacitors in series chained negative plate to positive plate, such that each plate holds the same charge, and the charge on the combination is the same as on each capacitor However, the potential difference delivered by the battery across the equivalent capacitance C s is imparted across the capacitors in series. Hence V V V 1 2 Q Q Q Cs C1 C2 1 1 1 CC 1 2 Cs C C C C C s 1 2 1 2 For n capacitors in series: 1 1 1 1... C C C C s Q Q Q 1 2 1 2 n Notice that the series equivalent capacitance is smaller than any of the individual capacitances

Problems: 10. Mixed combinations of capacitors: A capacitive circuit combines capacitors as in the figure (the numbers are capacitances in μf). a) Find the equivalent capacitance of the capacitive circuit in the figure. b) Say that a 12-V battery is connected between points ab. What is the amount of charge stored on the combination of capacitors? 11. Capacitive circuit analysis: Three capacitors are connected across a 12-V battery as on the figure. a) Find the equivalent capacitance of the circuit b) Find the charge on each capacitor in the circuit and the potential difference across it

Energy Stored in a Capacitor The energy stored in a capacitor is equal to the energy necessary to increase the charge on the plates from zero to Q: Q 1 W U Vdq qdq C Q 0 0 U 1 2 1 2 2 2 Q 2C From the definition of capacitance, this can be rewritten in different forms U CV QV q Q Q V Therefore, we see that a capacitor can be seen a charge or energy storage device When connected across a conductive medium this energy is released. In general, capacitors act as energy reservoirs that can slowly charged and then discharged quickly to provide large amounts of energy in a short pulse We can define the energy density u as the energy per unit volume. For a parallel plate capacitor (but with a result valid for any capacitor) 1 A 2 1 2 2 0 Ed U 2 CV d u Volume Ad Ad u Vacuum 1 2 E 0 2

Dielectrics Capacitors with dielectrics A dielectric is an insulating material that, when placed between the plates of a capacitor, increases the capacitance Ex: Dielectrics can be rubber, plastic, or waxed paper If the capacitance of a capacitor with air between the plates is C 0, when a dielectric completely fills the region between the plates, the capacitance increases by the factor κ > 1 called dielectric constant: C C A A 0 0 d d ε =κε 0 is called the electric permittivity of the dielectric Ex: A dielectric improves the performance of a capacitor: a) Say that a capacitor without dielectric stores a certain amount of charge Q 0. The voltage across the plates is V 0. b) With the dielectric inside, the charge stays the same but C increases and V decreases: the same charge is held with a lower V. Dielectric E0 1 E u 2 E Quiz 6: Is the energy increasing or decreasing when a dielectric is inserted in between the charged plates? a) Increases b) Decreases c) Stays the same 2

Dielectrics Polarization If we decrease V we also decrease E which is done by a polarization in the dielectric resulting in an electric field opposite to the initial field. Polarization occurs when there is a separation between the centers of gravity of negative and positive charge of the molecules In a capacitor, the dielectric becomes polarized because it is in an electric field that exists between the plates For any given plate separation, there is a maximum electric field that can be produced in the dielectric before it breaks down and begins to conduct This maximum electric field is called the dielectric strength The polarization results in an induced surface charge density σ i which decreases the net charge density σ σ i E 1 1 i 0 E i 0 0 Therefore, for a high κ dielectric, the induced density can almost cancel out the density on the plates, so a small potential difference will hold a large charge density σ on the plates

Problems: 12. Dielectrics in series and parallel: A parallel plate capacitor of capacitance C 0 has the space between the plates filled with two slabs of dielectric, with constants κ 1 and κ 2. What is the capacitance in terms of C 0, d, κ 1 and κ 2 when the space is filled as in figure a) and then as in figure b): a) b) d κ 1 κ 2 ½d d κ 1 κ 2 13. Energy in capacitors: Three parallel-plate capacitors are networked as in the figure with given geometrical characteristics. Together they hold a net charge Q. A particle of mass m and charge q hangs above a very long wire of static charge density λ. The particle is released from rest and pulls the dielectric out of a capacitor. Sketch the steps necessary to calculate the speed of the particle at the moment when the dielectric is completely out, using conservation of energy.