Chapter 16: Electric Energy and Capacitance Potential Difference and Electric Potential  Work and electric potential energy Consider a small positive charge placed in a uniform electric field E d A B conservative force the force is in the same direction as the net displacement of the test charge The work done by a conservative force can be reinterpreted as the negative of the change in a potential energy associated with that force. SI unit : joule (J) Homework assignment: 9,18,28,34,44,51 (Six problems!)

Electric Potential Energy In analogy to the gravitational force, a potential can be defined as: When the test charge moves from height y A to height y B, the work done on the charge by the field is given by: U increases (decreases) if the test charge moves in the direction opposite to (the same direction as) the electric force + + A B  PE<0 + + B A  PE>0 - - A B - - B A  PE<0  Work and electric potential energy (cont’d)

Electric Potential Energy In analogy to the gravitational force, a potential can be defined as: When the test charge moves from height y A to height y B, the work done on the charge by the field is given by:  Work and electric potential energy (cont’d) Define an electric potential difference as: Then for a special case of a uniform electric field: SI unit : joule per coulomb or volt (J/C or V) SI unit : J/C=V=N/C The change in electric potential energy as a charge q moves from A to B divided by charge q.

Electric Potential Energy  Example 16.1 : Potential energy differences in an electric field A proton is released from rest at x=-2.00 cm in a constant electric field with magnitude 1.50x10 3 V/C pointing the positive x-direction. (a)Calculate the change in the electric potential energy associated with the proton when it reaches x=5.00 cm. (b) Find the change in electric potential energy associated with an electron fired from x=-2.00 cm and reaching x=12.0 m. (b) Find the change in electric potential energy associated with an electron fired from x=3.00 cm to x=7.00 cm if the direction of the electric field is reversed.

Electric Potential Energy  Example 16.2 : Dynamics of charged particles Continuation of Example (a)Find the speed of the proton at x= m in part (a) of Example (b) Find the electron’s initial speed, given that its speed has fallen by half at x=0.120 m.

Electric Potential Energy  Example 16.3 : TV tubes and atom smasher Charged particles are accelerated through potential difference. Suppose a proton is injected at a speed of 1.00x10 6 m/s between two plates 5.00 cm apart. The proton subsequently accelerates across the gap and exits through the opening. (a)What must the electric potential difference be if the exist speed is to be 3.00x10 6 m/s? (b) What is the magnitude of the electric field between the plates?

Electric Potential and Potential Energy  Electric potential by a point charge The electric field of a point charge extends throughout space, so does its electric potential. The zero point can be anywhere but it is convenient to choose the point at infinity. Then it can be shown that the electric potential due to a point charge q at a distance r is given by : The superposition principle : the total electric potential at some point P due to several point charges is the algebraic sum of the electric potentials due to the individual charges.

Electric Potential and Potential Energy  Electric potential energy of a pair of point charges If V 1 is the electric potential due to charge q 1 at a point P, then the work required to bring charge q 2 from infinity to P without acceleration is q 2 V 1. This work is, by definition, equal to the potential energy PE of the two-particle system when the particles are separated by a distance r. If q 1 q 2 >0, PE>0 If q 1 q 2 <0, PE<0  Electric potential energy with several point charges From the superposition principle

Electric Potential and Potential Energy  Example 16.4 : Finding the electric potential A  C point charge is at origin and a point charge q 2 =-2.00  C is on the x-axis at (3.00,0) m. +- P (0,4.00) (3.00,0) y(m) x(m) q1q1 q2q2 0 r2r2 r1r1 (a)Find the electric potential at point P due to these charges. (b) Find the work needed to bring the  C charge from infinity to P.

Potentials and Charged Conductors  Surface electric potential of a conductor in electrostatic equilibrium Work done to move a charge from point A to point B No net work is needed to move a charge between two points that are at the same electric potential. We will learn that when a charged conductor is in electrostatic equilibrium: -A net charge placed on it resides entirely on its surface. -The electric field just outside its surface is perpendicular to the surface and that the field inside the conductor is zero. -All points on its surface are at the same potential.

 Equipotential surface E = 0 everywhere inside a conductor - At any point just inside the conductor the component of E tangent to the surface is zero - The tangential component of E is also zero just outside the surface When all charges are at rest, the surface of a conductor is always an equipotential surface conductor vacuum If it were not, a charge could move around a rectangular path partly inside and partly outside and return to its starting point with a net amount of work done on it. When all charges are at rest, the electric field just outside a conductor must be perpendicular to the surface at every point Potentials and Charged Conductors

 Examples of equipotential surface Potentials and Charged Conductors The equipotentials are perpendicular to the electric field lines at every point.

 Electron volt Potentials and Charged Conductors The electron volt is defined as the kinetic energy that an electron gains when accelerated through a potential difference of 1V. 1 V = 1 J/C 1 eV = 1.60 x C. V = 1.60 x J In atomic, nuclear and particle physics, the electron volt is used commonly to express energies. - Electrons in normal atoms have energies of tens of eV’s. - Excited electrons in atoms that emit x-rays have energies of thousands of eV’s ( keV = 10 3 eV). - High energy gamma rays emitted by the nucleus have energies of millions of eV’s (MeV = 10 6 eV). - The world most energetic accelerator near Chicago accelerates protons/anti-proton up to Tera eV’s (TeV = eV )

Capacitance  Capacitor Any two conductors separated by an insulator (or a vacuum) form a capacitor In practice each conductor initially has zero net charge and electrons are transferred from one conductor to the other (charging the conductor) Then two conductors have charge with equal magnitude and opposite sign, although the net charge is still zero When a capacitor has or stores charge Q, the conductor with the higher potential has charge +Q and the other -Q if Q>0

Capacitors and Capacitance  Capacitance One way to charge a capacitor is to connect these conductors to opposite terminals of a battery, which gives a fixed potential difference V ab between conductors ( a-side for positive charge and b-side for negative charge). Then once the charge Q and –Q are established, the battery is disconnected. If the magnitude of the charge Q is doubled, the electric field becomes twice stronger and V ab =  V is twice larger. Then the ratio Q  V is still constant and it is called the capacitance C. When a capacitor has or stores charge Q, the conductor with the higher potential has charge +Q and the other -Q if Q>0 -Q Q a b

 Parallel-plate capacitor in vacuum Charge density: Electric field: Potential diff.: Capacitance: The capacitance depends only on the geometry of the capacitor. It is proportional to the area A. It is inversely proportional to the separation d When matter is present between the plates, its properties affect the capacitance. Parallel-Plate Capacitance

 Units 1 F = 1 C 2 /N m (Note [    C 2 /N m 2 )  0 = 8.85 x F/m 1  F = F, 1 pF = F  Example : Size of a 1-F capacitor

 Example : Properties of a parallel capacitor Parallel-Plate Capacitance

Combinations of Capacitors  Symbols for circuit elements and circuits

 Capacitors in parallel a b The parallel combination is equivalent to a single capacitor with the same total charge Q=Q 1 +Q 2 and potential difference. Combinations of Capacitors

 Capacitors in series a b c The equivalent capacitance C eq of the series combination is defined as the capacitance of a single capacitor for which the charge Q is the same as for the combination, when the potential difference V is the same. Combinations of Capacitors

 Capacitor networks Combinations of Capacitors

 Capacitor networks (cont’d) Combinations of Capacitors

 Capacitor networks 2 C A B A B CC C CC CC C CC C C C C A B CC C C C A B Combinations of Capacitors

Energy Stored in a Charged Capacitor  Work done to charge a capacitor Consider a process to charge a capacitor up to Q with the final potential difference  V. Let q i and (  v) i be the charge and potential difference at an intermediate stage during the charging process. At this stage the work (  W) i required to transfer an additional element of charge  q is: The total work needed to increase the capacitor charge q from zero to Q is: Q VV qq  v) i qiqi The energy stored:

Capacitor with Dielectrics  Dielectric materials Experimentally it is found that when a non-conducting material (dielectrics) between the conducting plates of a capacitor, the capacitance increases for the same stored charge Q. Define the dielectric constant  as: When the charge is constant, Material  vacuum 1 air(1 atm) Teflon 2.1 Polyethelene 2.25 Mica 3-6 Mylar 3.1 Plexiglas 3.40 Water 80.4 C 0 : capacitance w/o dielectric C : capacitance w/ dielectric

Dielectrics  Induced charge and polarization Consider a two oppositely charged parallel plates with vacuum between the plates. Now insert a dielectric material of dielectric constant . Source of change in the electric field is redistribution of positive and negative charge within the dielectric material (net charge 0). This redistribution is called a polarization and it produces induced charge and field that partially cancels the original electric field.

Dielectrics  Molecular model of induced charge

Dielectrics  Molecular model of induced charge (cont’d)