Electrical Potential Energy In Chapter 15, we saw that the gravitational and electrical (Coulomb) forces have similar forms Fg G m1m2 r2 Gravity Fe k e q1 q2 r2 Electrical This similarity also leads to a similarity between the potential energies associated with each force

U g G m1m2 r q1q2 U e k e r (can be obtained directly through calculus) Gravity Electrical Ue depends on magnitude and sign of a pair of charges Ue is positive (negative) when q1 and q2 have the same (opposite) sign Remember: potential energy is a scalar quantity

Electrical Potential Energy Comparison of Ug and Ue as a function of separation U U U distance: r r r g e e q1q2 < 0 If 2 charges have opposite (same) signs, the potential energy of the pair increases

(decreases) with separation distance Charges always move from high to low potential energy Positive (negative) charges move in the same (opposite) direction as the electric field q1q2 > 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? A) B) C)

D) 10 m/s 14 m/s 20 m/s Since 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 Potential Electric potential is defined as the electric potential energy per unit charge Ue V q Scalar quantity with units of volts (1 V = 1 J/C)

Sometimes called simply potential or voltage Electric potential is characteristic of the field only, independent of a test charge placed in that field Potential energy is a characteristic of a charge-field system due to an interaction between the field and a charge placed in the field When 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 potential Lower potential Electric Potential When a point charge q moves between 2 points A

and B, it moves through a potential difference: V V f Vi VB VA The potential difference is the change in electric potential energy per unit charge: U e qV 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) potential Helpful detail: E points in the direction of decreasing V Electric potential created by a point charge: Depends only on q and r Potential vs. Potential Energy V ke

q r Example Problem #16.17 The 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. 3.87 cm 2 1 Solution (details given in class): 11.0 kV

P 3 Potential Differences in Biological Systems Axons (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 ms This changes polarity of inside fluid to + Potential difference across cell membrane changes from about 85 mV to +60 mV Restoration 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 Medicine Polarity changes across membranes of muscle cells Muscle cells have a layer of ions on the inside of the membrane and + ions on the outside Just before each heartbeat, + ions are pumped into the cells, neutralizing the potential difference (depolarization) Cells become polarized again when the heart relaxes Electrocardiogram (EKG) Measures potential difference between points on chest as a function of time Polarization and depolarization of cells in heart causes potential differences that are measured by electrodes Electroencephalogram (EEG) and Electroretinogram (ERG) Measures potential differences caused by electrical activity in the brain (EEG) and retina (ERG)

Potentials and Charged Conductors We know that: U = W (from last semester) and U = qV Combining these two equations: W qV qVB VA No work is required to move a charge between two points at the same electric potential For a charged conductor in equilibrium: No work is done by E if charge is moved between points A and B Since W = 0, VB VA = 0 at surface Since E = 0 inside a conductor, no work is required to move a charge inside conductor (thus V = 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? A) B) C) D) 1V 10 V 100 V 1000 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) A) B) C) D) 0.8 J 8J 12 J 1200 J Equipotential Surfaces An equipotential surface has the same potential at every point on the surface Similar to topographic map, which shows lines of constant elevation Since V = 0 for each surface, W = 0

along the surface Thus electric field lines are perpendicular to the equipotential surfaces at all points E points in the direction of the maximum decrease in V (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 Surfaces On a contour map a hill is steepest where the lines of constant elevation are close together If 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 Which equipotential plot best represents the electric field pattern shown? A) B) C) D) Plot 1 Plot 2 Plot 3 Plot 4 (PHYSLET Physics Exploration 25.1, copyright Pearson Prentice Hall, 2004) Capacitance

A capacitor is a device that stores electrical potential energy by storing separated + and charges 2 conductors separated by vacuum, air, or insulation + charge put on one conductor, equal amount of charge put on the other conductor A battery or power supply typically supplies the work necessary to separate the charge Simplest form of capacitor is the Charging A Capacitor parallel plate capacitor 2 parallel plates, each with same area A, separated by distance d Charge +Q on one plate, Q on the other If plates are close together, electric field will be uniform (constant) between the plates Capacitance For a uniform electric field, the potential difference

between the plates is (see Example Problem #16.6) V = Ed E is proportional to the charge, and V is proportional to E therefore the charge is proportional to V The constant of proportionality between charge and V is called capacitance Q C V Units: C / V = Farad (F) Capacity to hold charge for a given V 1 F is very large unit: typical values of C are F, nF, or pF Capacitance depends on the geometry of the plates and the material between the plates

C 0 A (for plates separated by air) d Capacitors in Circuits and Applications Capacitors are used in a variety of electronic circuits Example of circuit diagram consisting of capacitors and a battery shown at right Many practical uses of capacitors Some computer keyboard keys have capacitors with a variable plate spacing below them Microphones using capacitors with one moving plate to create an electrical signal Constant potential difference kept between plates by a battery As plate spacing changes, charge flows onto and off of plates

The moving charge (current) is amplified to generate signal Tweeters (speakers for high-frequency sounds) are microphones in reverse Millions of microscopic capacitors used in each RAM computer memory chip Charged and discharged capacitors correspond to 1 and 0 states CQ 5: Interactive Example Problem: Fun With Capacitors If a constant electric potential is maintained between the plates of the capacitor, what happens to the charge on the capacitor? A) B) C) D) The charge gets smaller.

The charge gets larger. The charge stays the same. The capacitor discharges. (PHYSLET Physics Exploration 26.2, copyright Pearson Prentice Hall, 2004) Combinations of Capacitors Capacitors can be combined in circuits to give a particular net capacitance for the entire circuit Parallel Combination Potential difference across each capacitor is the same and equal to V of the battery Qtot = Q1 + Q2 + Q3 + Total (equivalent) capacitance: Ceq C1 C2 C3 Series Combination Magnitude of charge is the same on

all plates V (battery) = V1 + V2 + V3 + Total (equivalent) 1 1 1 1 capacitance: Ceq C1 C2 C3 Example Problem Capacitors C1 = 4.0 F and C2 = 2.0 F 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. Solution (details given in class): 1.8 102 C (4.0 F capacitor) 89 C (2.0 F capacitor)

Example Problem #16.35 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. Solution (details given in class): (a) 2.67 F (b) 24.0 C (each 8.00-F capacitor), 18.0 C (6.00-F capacitor), 6.00 C (2.00-F capacitor) (c) 3.00 V (each capacitor) Energy Stored in a Charged Capacitor Its 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 (W = VQ) Total work done = total energy stored: 1 1 Q2 2 E QV C V 2 2 2C Defibrillators typically release about 1.2 kJ of stored energy from capacitor with V 5 kV Capacitors with Dielectrics A dielectric is an insulating material Rubber, plastic, glass, nylon

When a dielectric is inserted between the conductors of a capacitor, the capacitance increases Capacitance increases for a parallel-plate capacitor in which a dielectric fills the entire space between the plates A = dielectric constant (ratio of capacitance with dielectric to capacitance without dielectric) C 0 d 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 conducts See Table 16.1 for values of and dielectric strength for various materials

Capacitors with Dielectrics The molecules of the dielectric, when placed in the electric field of a capacitor, become polarized Centers 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 V Since plates can store more charge for a given voltage, the capacitance must increase (remember C = Q / V ) Capacitors with Dielectrics To increase capacitance while keeping the physical size reasonable, plates are often made of a thin conducting foil that is rolled into a cylinder Dielectric material is sandwiched in between High-voltage capacitor commonly consists of

interwoven metal plates immersed in silicone oil Very large capacitances can be achieved with an electrolytic capacitor at relatively low voltages Insulating metal oxide layer forms on the conducting foil and serves as a (very thin) dielectric