Electric Potential and Capacitance What’s a volt anyway? Presentation 2001 Dr. Phil Dauber as modified by R. McDermott

Why Potential? Electric potential can be visualized as “height”. Electric potential can be visualized as “height”. This allows us to make comparisons to gravity. This allows us to make comparisons to gravity. Potential is independent of the object in the electric field. Potential is independent of the object in the electric field.

What is Potential? By agreement from fields, we always define electrical quantities in terms of positive charge. By agreement from fields, we always define electrical quantities in terms of positive charge. Positive charges will always move away from other positive charges and toward negative charges, much as a ball always rolls downhill. Positive charges will always move away from other positive charges and toward negative charges, much as a ball always rolls downhill. By analogy, we consider a region of positive charge to be at a high potential (hill), and a region of negative charge to be at a low potential (valley). By analogy, we consider a region of positive charge to be at a high potential (hill), and a region of negative charge to be at a low potential (valley). Field lines, then, show the most direct “downhill” path. Field lines, then, show the most direct “downhill” path.

Equipotential Lines Maps dealing with hills and valleys are called contour or topographical maps, and consist of closed lines that connect points that have the same height or altitude. Maps dealing with hills and valleys are called contour or topographical maps, and consist of closed lines that connect points that have the same height or altitude. We draw similar “maps” for dealing with charges using Equipotential lines, which connect points that have the same electrical height (potential). We draw similar “maps” for dealing with charges using Equipotential lines, which connect points that have the same electrical height (potential). An equipotential line will always be perpendicular to the electric field, since the field always points “downhill”. An equipotential line will always be perpendicular to the electric field, since the field always points “downhill”. Stationary charges will not move on their own along an equipotential line, because that will not lead them “downhill”. Stationary charges will not move on their own along an equipotential line, because that will not lead them “downhill”.

Hills and Valleys, Oh My! As a ball rolls downhill, the gravitational field does work on the ball due to the unbalanced force (weight) acting on the ball, causing it to gain speed (and kinetic energy). We say that potential energy stored in the field between the ball and Earth is converted into kinetic energy. As a ball rolls downhill, the gravitational field does work on the ball due to the unbalanced force (weight) acting on the ball, causing it to gain speed (and kinetic energy). We say that potential energy stored in the field between the ball and Earth is converted into kinetic energy. Similarly, as a positive charge moves from high potential to low the electric field does work on the charge due to the unbalanced force acting on it, causing it to gain speed (and kinetic energy). We say that potential energy stored by the field and charge is converted into kinetic energy. Similarly, as a positive charge moves from high potential to low the electric field does work on the charge due to the unbalanced force acting on it, causing it to gain speed (and kinetic energy). We say that potential energy stored by the field and charge is converted into kinetic energy.

Hills and Valleys, Oh My!

Electric and Gravitational Fields Property: mass Property: mass Property: charge Property: charge Gravity: Electricity: 1 Sign: positive 1 Sign: positive Dependency: 1/r 2 Dependency: 1/r 2 2 Signs: pos, neg 2 Signs: pos, neg k = 8.99x10 9 n-m 2 /C 2 k = 8.99x10 9 n-m 2 /C 2 G = 6.67x n-m 2 /kg 2 G = 6.67x n-m 2 /kg 2

Electric and Gravitational Fields Gravity: Electricity: F = (kq 1 /r 2 ) q 2 F = (kq 1 /r 2 ) q 2 F = E field q 2 F = E field q 2 F= (Gm Earth /r Earth 2 ) m 2 F= (Gm Earth /r Earth 2 ) m 2 F = g field m 2 F = g field m 2 PE = mg  h PE = mg  h PE = qE  x (  x = d) PE = qE  x (  x = d) PE is in Joules PE is in Joules

Electric and Gravitational Fields Gravity: Electricity: We want to treat all objects the same way, so.. We want to treat all objects the same way, so.. “ Potential”, V = Eqd/q = Ed “ Potential”, V = Eqd/q = Ed V = (PE)/charge V = (PE)/charge Units = Joules/Coul. Units = Joules/Coul. L= (PE)/mass L= (PE)/mass Units = Joules/kg Units = Joules/kg We want to treat all objects the same way, so.. We want to treat all objects the same way, so.. “Liftage”, L = mg  h/m = g  h “Liftage”, L = mg  h/m = g  h

Electric and Gravitational Fields Gravity: Electricity: E = F/q E = F/q Units = N/c Units = N/c E is the electric field strength E is the electric field strength g = F/m g = F/m Units = N/kg Units = N/kg g is the grav. field strength g is the grav. field strength

Electric Potential and Electric Field Can describe charge distribution in terms field or potential. Consider uniform field: Can describe charge distribution in terms field or potential. Consider uniform field: F = Eq F = Eq W = qV ba W = qV ba W = Fd = qE d W = Fd = qE d Thus V ba = Ed or E = V ba /d Thus V ba = Ed or E = V ba /d Alternate units for E: volts per coulomb Alternate units for E: volts per coulomb

Electrical Potential Potential is potential energy per unit charge Potential is potential energy per unit charge Analogous to field which is force per unit charge Analogous to field which is force per unit charge Symbol of potential is V; V a = PE a /q Symbol of potential is V; V a = PE a /q Only potential differences are measurable; zero point of potential is arbitrary Only potential differences are measurable; zero point of potential is arbitrary V ab = V a – V b = - W ba /q V ab = V a – V b = - W ba /q W ba is work done to move q from b to a W ba is work done to move q from b to a

Units: Unit of electric potential is the volt Unit of electric potential is the volt Abbreviation V Abbreviation V 1 V = 1 Joule/Coulomb = 1 J/C 1 V = 1 Joule/Coulomb = 1 J/C Thus electrical work = qV If q is in coulombs, the work is in joules; if q is in elementary charges, the work is in electron-volts (eV). Thus electrical work = qV If q is in coulombs, the work is in joules; if q is in elementary charges, the work is in electron-volts (eV). Potential difference is called voltage Potential difference is called voltage

Electrical Potential Energy The change in potential energy equals the negative of the work done by the field. The change in potential energy equals the negative of the work done by the field. Electrical PE is transferred to the charge as kinetic energy. Electrical PE is transferred to the charge as kinetic energy. A positive charge has its greatest PE near another positive charge or positive plate. A positive charge has its greatest PE near another positive charge or positive plate. Only differences in potential energy are measurable Only differences in potential energy are measurable

V = 0 Arbitrary Usually ground is zero point of potential Usually ground is zero point of potential Sometimes potential is zero at infinity Sometimes potential is zero at infinity + terminal of 12V battery is + terminal of 12V battery is said to be at 12V higher said to be at 12V higher potential than – terminal potential than – terminal

Electric Potential and Potential Energy   PE = PE b – PE a = qV ba  If object with charge q moves through a potential difference V ba its potential energy changes by qV ba  Example: What is the gain of electrical PE when 1 C of charge moves between the terminals of a 12 volt battery? Water Analogy: Water Analogy: Voltage is like water pressure (depth or height) Voltage is like water pressure (depth or height)

Example: Electron in Computer Monitor An electron is accelerated from rest through a potential difference of 5000 volts An electron is accelerated from rest through a potential difference of 5000 volts Find its change in potential energy Find its change in potential energy Find its speed after acceleration Find its speed after acceleration a b V ba = 5000 v = V b - V a  PE = 8x J V = 4.2x10 7 m/s

Questions on Preceding Example Does the energy depend on the particle’s mass? Does the energy depend on the particle’s mass? Does the final speed depend on the mass? Does the final speed depend on the mass?

Electric Potential Due to Point Charges V = k Q/r derived from Calculus V = k Q/r derived from Calculus Here V = 0 at r = infinity; V represents potential difference between r and infinity Here V = 0 at r = infinity; V represents potential difference between r and infinity

Example: Work to force two point charges together What work is needed to bring a 2  C charge to a point 10cm from a 5  c charge? What work is needed to bring a 2  C charge to a point 10cm from a 5  c charge? Work required = change in potential energy Work required = change in potential energy W = qV ba = q{kQ/r b – kQ/r a } W = qV ba = q{kQ/r b – kQ/r a }

The Electron Volt The energy acquired by a particle carrying a charge equal to that of the electron when accelerated through one volt. The energy acquired by a particle carrying a charge equal to that of the electron when accelerated through one volt. W = qV, if q is in Coulombs, then W is in Joules. If q is in elementary charges, W is in electron volts (eV). W = qV, if q is in Coulombs, then W is in Joules. If q is in elementary charges, W is in electron volts (eV). 1 eV = 1.6 x joules 1 eV = 1.6 x joules It’s about ENERGY! It’s about ENERGY! 1 KeV = 1000 eV; 1 MeV = 10 6 eV 1 KeV = 1000 eV; 1 MeV = 10 6 eV 1 Gev = 10 9 eV 1 Gev = 10 9 eV

Potential at an Arbitrary Point Near Several Point Charges Add the potentials due to each point charge Add the potentials due to each point charge Use the right sign for the charge Use the right sign for the charge Relax; potential isn’t a vector Relax; potential isn’t a vector What is true about the mid-plane between two equal point charges of opposite sign? What is true about the mid-plane between two equal point charges of opposite sign?

Capacitors Store charge Store charge Two conducting plates Two conducting plates NOT touching NOT touching May have insulating May have insulating material between material between Q = CV Q = CV

Capacitance Symbol C Symbol C Unit: coulombs per volt = farad Unit: coulombs per volt = farad 1 pF = 1 picofarad = farad 1 pF = 1 picofarad = farad 1nf = 1 nanofarad = f 1nf = 1 nanofarad = f 1  f = 1 microfarad = 1 -6 f 1  f = 1 microfarad = 1 -6 f

C is Constant for a Given Capacitor Does not depend on Q or V Does not depend on Q or V Proportional to area Proportional to area Inversely proportional to distance between plates Inversely proportional to distance between plates C =  0 A/d C =  0 A/d If dielectic like oil or paper between plates use e = K  0 ; K is called dielectric constant If dielectic like oil or paper between plates use e = K  0 ; K is called dielectric constant

Find the Capacitance A capacitor can hold 5  C of charge at a potential difference of 100 volts. What is its capacitance A capacitor can hold 5  C of charge at a potential difference of 100 volts. What is its capacitance C = Q/V = 5 x C/ 100V = C = Q/V = 5 x C/ 100V = 5 x f = pf 5 x f = pf

Capacitors Photos courtesy Illinois Capacitor, Inc

Applications In automotive ignitions In automotive ignitions In strobe lights In strobe lights In electronic flash In electronic flash In power supplies In power supplies In nearly all electronics In nearly all electronics

A Capacitor Stores Electric Energy A battery produces electric energy bit by bit A battery produces electric energy bit by bit A capacitor is NOT a type of battery A capacitor is NOT a type of battery A battery can be used to charge a capacitor A battery can be used to charge a capacitor

Energy in a Capacitor Holding Charge Q at Voltage V U = QV/2 = CV 2 /2 = Q 2 /2C U = QV/2 = CV 2 /2 = Q 2 /2C Derivation: the work needed to charge a capacitor by bringing charge onto a plate when some is already there (use W =QV) Derivation: the work needed to charge a capacitor by bringing charge onto a plate when some is already there (use W =QV) Initially V = 0 Initially V = 0 Average voltage during the charging process is V/2 Average voltage during the charging process is V/2