Electrical Potential EnergyIn Chapter 15, we saw that the gravitational and electrical (Coulomb) forces have similar forms

This similarity also leads to a similarity between the potential energies associated with each force

Ue depends on magnitude and sign of a pair of charges Ue is positive (negative) when q1 and q2 have the same (opposite) signRemember: potential energy is a scalar quantityGravityElectricalGravityElectrical(can be obtained directly through calculus)

Electrical Potential EnergyComparison of Ug and Ue as a function of separation distance:

If 2 charges have opposite (same) signs, the potential energy of the pair increases (decreases) with separation distanceCharges always move from high to low potential energyPositive (negative) charges move in the same (opposite) direction as the electric field UgrUerUerq1q2 < 0q1q2 > 0

CQ 1: A positively charged particle starts at rest 25 cm from a second positively charged particle which is held stationary throughout the experiment. The first particle is released and accelerates directly away from the second particle. When the first particle has moved 25 cm, it has reached a velocity of 10 m/s. What is the maximum velocity that the first particle will reach? 10 m/s 14 m/s20 m/sSince the first particle will never escape the electric field of the second particle, it will never stop accelerating, and will reach an infinite velocity.

Electric PotentialElectric potential is defined as the electric potential energy per unit chargeScalar quantity with units of volts (1 V = 1 J/C)Sometimes called simply potential or voltageElectric potential is characteristic of the field only, independent of a test charge placed in that fieldPotential energy is a characteristic of a charge-field system due to an interaction between the field and a charge placed in the fieldWhen a positive (negative) charge is placed in an electric field, it moves from a point of high (low) potential to point of lower (higher) potential Higher potentialLower potential

Electric PotentialWhen a point charge q moves between 2 points A and B, it moves through a potential difference:

The potential difference is the change in electric potential energy per unit charge:The electric force on any charge (+ or ) is always directed toward regions of lower electric potential energy (just like gravity)For a positive (negative) charge, lower potential energy means lower (higher) potentialHelpful detail: E points in the direction of decreasing V Electric potential created by a point charge:Depends only on q and rPotential vs. Potential Energy

Example Problem #16.17Solution (details given in class):11.0 kVThe three charges shown in the figure are at the vertices of an isosceles triangle. Let q = 7.00 nC, and calculate the electric potential at the midpoint of the base.

Potential Differences in Biological SystemsAxons (long extensions) of nerve cells (neurons)In resting state, fluid inside has a potential that is 85 mV relative to the fluid outside (due to differences in +/ ion concentrations)A nerve impulse causes the outer membrane to become permeable to + Na ions for about 0.2 msThis changes polarity of inside fluid to +Potential difference across cell membrane changes from about 85 mV to +60 mVRestoration of resting potential involves the diffusion of K and pumping of Na ions out of cell (active transport)As much as 20% of the resting energy requirements of the body are used for active transport of Na ions

Potential Differences in MedicinePolarity changes across membranes of muscle cellsMuscle cells have a layer of ions on the inside of the membrane and + ions on the outsideJust before each heartbeat, + ions are pumped into the cells, neutralizing the potential difference (depolarization)Cells become polarized again when the heart relaxesElectrocardiogram (EKG)Measures potential difference between points on chest as a function of timePolarization and depolarization of cells in heart causes potential differences that are measured by electrodesElectroencephalogram (EEG) and Electroretinogram (ERG)Measures potential differences caused by electrical activity in the brain (EEG) and retina (ERG)

Potentials and Charged ConductorsWe know that: DU = W (from last semester) and DU = qDV Combining these two equations:No work is required to move a charge between two points at the same electric potentialFor a charged conductor in equilibrium:No work is done by E if charge is moved between points A and BSince W = 0, VB VA = 0 at surfaceSince E = 0 inside a conductor, no work is required to move a charge inside conductor (thus DV = 0 inside as well)Conclusion: Electric potential is constant everywhere inside a conductor and is equal to its (constant) value at the surface

CQ 2: Two charged metal plates are placed one meter apart creating a constant electric field between them. A one Coulomb charged particle is placed in the space between them. The particle experiences a force of 100 Newtons due to the electric field. What is the potential difference between the plates? 1 V 10 V100 V1000 V

CQ 3: How much work is required to move a positively charged particle along the 15 cm path shown, if the electric field E is 10 N/C and the charge on the particle is 8 C? (Note: ignore gravity) 0.8 J8 J12 J1200 J

Equipotential SurfacesAn equipotential surface has the same potential at every point on the surfaceSimilar to topographic map, which shows lines of constant elevationSince DV = 0 for each surface, W = 0 along the surfaceThus electric field lines are perpendicular to the equipotential surfaces at all points E points in the direction of the maximum decrease in DV (E points from high to low potential)Similar to a topographic contour map (slope is steepest perpendicular to lines of constant elevation)Electric field is thus sometimes called the potential gradient (meaning grade or slope)

Equipotential SurfacesOn a contour map a hill is steepest where the lines of constant elevation are close togetherIf equipotential surfaces are drawn such that the potential difference between adjacent surfaces is constant, then the surfaces are closer together where the field is stronger

Examples of Equipotential Surfaces

CQ 4: Interactive Example Problem:Drawing Equipotential Lines(PHYSLET Physics Exploration 25.1, copyright Pearson Prentice Hall, 2004)Which equipotential plot best represents the electric field pattern shown?Plot 1Plot 2Plot 3Plot 4

CapacitanceA capacitor is a device that stores electrical potential energy by storing separated + and charges2 conductors separated by vacuum, air, or insulation+ charge put on one conductor, equal amount of charge put on the other conductorA battery or power supply typically supplies the work necessary to separate the chargeSimplest form of capacitor is the parallel plate capacitor2 parallel plates, each with same area A, separated by distance dCharge +Q on one plate, Q on the otherIf plates are close together, electric field will be uniform (constant) between the platesCharging A Capacitor

CapacitanceFor a uniform electric field, the potential difference between the plates is (see Example Problem #16.6) DV = Ed E is proportional to the charge, and DV is proportional to E therefore the charge is proportional to DVThe constant of proportionality between charge and DV is called capacitance

Capacity to hold charge for a given DV 1 F is very large unit: typical values of C are mF, nF, or pFCapacitance depends on the geometry of the plates and the material between the platesUnits: C / V = Farad (F)(for plates separated by air)

Capacitors in Circuits and ApplicationsCapacitors are used in a variety of electronic circuitsExample of circuit diagram consisting of capacitors and a battery shown at rightMany practical uses of capacitorsSome computer keyboard keys have capacitors with a variable plate spacing below themMicrophones using capacitors with one moving plate to create an electrical signalConstant potential difference kept between plates by a batteryAs plate spacing changes, charge flows onto and off of platesThe moving charge (current) is amplified to generate signalTweeters (speakers for high-frequency sounds) are microphones in reverseMillions of microscopic capacitors used in each RAM computer memory chipCharged and discharged capacitors correspond to 1 and 0 states

CQ 5: Interactive Example Problem:Fun With Capacitors(PHYSLET Physics Exploration 26.2, copyright Pearson Prentice Hall, 2004)If a constant electric potential is maintained between the plates of the capacitor, what happens to the charge on the capacitor?The charge gets smaller.The charge gets larger.The charge stays the same.The capacitor discharges.

Combinations of CapacitorsCapacitors can be combined in circuits to give a particular net capacitance for the entire circuitParallel CombinationPotential difference across each capacitor is the same and equal to DV of the battery Qtot = Q1 + Q2 + Q3 + Total (equivalent) capacitance:

Series CombinationMagnitude of charge is the same on all plates DV (battery) = DV1 + DV2 + DV3 + Total (equivalent) capacitance:

Example ProblemSolution (details given in class):1.8 102 mC (4.0 mF capacitor)89 mC (2.0 mF capacitor)Capacitors C1 = 4.0 mF and C2 = 2.0 mF are charged as a series combination across a 100V battery. The two capacitors are disconnected from the battery and from each other. They are then connected positive plate to positive plate and negative plate to negative plate. Calculate the resulting charge on each capacitor.

Example Problem #16.35Solution (details given in class):2.67 mF24.0 mC (each 8.00-mF capacitor), 18.0 mC (6.00-mF capacitor), 6.00 mC (2.00-mF capacitor)3.00 V (each capacitor)Find (a) the equivalent capacitance of the capacitors in the circuit shown, (b) the charge on each capacitor, and (c) the potential difference across each capacitor.

Energy Stored in a Charged CapacitorIts easy to tell that a capacitor stores (releases) energy when it charges (discharges)The energy stored by the capacitor = work required to charge the capacitor (typically performed by a battery or power supply)As more and more charge is transferred between the plates, the charge, voltage, and work done by battery increases (DW = DVDQ)Total work done = total energy stored:

Defibrillators typically release about 1.2 kJ of stored energy from capacitor with DV 5 kV

Capacitors with DielectricsA dielectric is an insulating material Rubber, plastic, glass, nylonWhen a dielectric is inserted between the conductors of a capacitor, the capacitance increasesCapacitance increases for a parallel-plate capacitor in which a dielectric fills the entire space between the plates k = dielectric constant (ratio of capacitance with dielectric to capacitance without dielectric)For any given plate separation d, there is a maximum electric field (dielectric strength) that can be produced in the dielectric before it breaks down and conductsSee Table 16.1 for values of k and dielectric strength for various materials

Capacitors with DielectricsThe molecules of the dielectric, when placed in the electric field of a capacitor, become polarizedCenters of positive and negative charges become preferentially oriented in the field (see figure below at left)Creates a net positive (negative) charge on the left (right) side of the dielectric (see figure below at right)This helps attract more charge to the conducting plates for a given DVSince plates can store more charge for a given voltage, the capacitance must increase (remember C = Q / DV )

Capacitors with DielectricsTo increase capacitance while keeping the physical size reasonable, plates are often made of a thin conducting foil that is rolled into a cylinderDielectric material is sandwiched in betweenHigh-voltage capacitor commonly consists of interwoven metal plates immersed in silicone oilVery large capacitances can be achieved with an electrolytic capacitor at relatively low voltagesInsulating metal oxide layer forms on the conducting foil and serves as a (very thin) dielectric