Unformatted text preview: Chapter 20
Electric Potential and Capacitance Electric Potential Energy
• When a point charge, qo, is placed in an electric field, it experiences a force • The work done by the electric field is • As this work is done by the field, the potential energy of the chargefield system is changed by U Electric Potential
• The potential energy per unit charge, U/qo, is the electric potential – The potential is independent of the value of qo – The potential has a value at every point in an electric field • As a charged particle moves in an electric field, it will experience a change in potential We often take the value of the potential to be zero at some convenient point in the field Sometimes called ground Potential and Potential Energy
• The potential is characteristic of the field only
– It is independent of the charge placed in the field (very small charge that does not alter the filed!) – The difference in potential is proportional to the difference in potential energy • Potential energy is characteristic of the charge+ field system
– Due to an interaction between the field and a charged particle placed in the field Work and Electric Potential
• The electric potential at an arbitrary point due to source charges equals the work required to bring a test charge from infinity to that point divided by the charge on the test particle – Assumes a charge moves slowly in an electric field without any change in its kinetic energy • The work performed on the charge is W= U=q V • 1 V = 1 J/C – V is a Volt – It takes one Joule of work to move a 1 Coulomb charge through a potential difference of 1 Volt • In addition, 1 N/C = 1 V/m  electric field – This indicates we can interpret the electric field as a measure of the rate of change with position of the electric potential • One electronvolt is defined as the energy a chargefield system gains or loses when a charge of magnitude e (an electron or a proton) is moved through a potential difference of 1 volt 1 eV = 1.60 x 1019 J Units Potential Difference in a Uniform Field
• The equations for electric potential can be simplified if the electric field is uniform: • The negative sign indicates that the electric potential at B is lower than at point A Energy and the Direction of Electric Field
• When the electric field is directed downward, point B is at a lower potential than point A • When a positive test charge moves from A to B, the chargefield system loses potential energy Electrical Potential Energy h d Gravitational Potential Energy Work = Fd = mgh G.P.E. = mgh Electrical Potential Energy Work = Fd = qEd E.P.E. = qEd Equipotentials
• Point B is at a lower potential than point A • Points B and C are at the same potential • The name equipotential surface is given to any surface consisting of a continuous distribution of points having the same electric potential Charged Particle in a Uniform Field, Example
• A positive charge is released from rest and moves in the direction of the electric field The change in potential is negative The change in potential energy is negative The force and acceleration are in the direction of the field • • • Potential and Point Charges
• A positive point charge produces a field directed radially outward • The potential difference between points A and B will be Potential and Point Charges, cont
• The electric potential is independent of the path between points A and B • It is customary to choose a reference potential of V = 0 at rA = • Then the potential at some point r is • Multiple charges Potential Energy of Multiple Charges
• Consider two charged particles • The potential energy of the system is •If there are more than two charges, then find U for each pair of charges and add them Potential of a Charge
• Now remove one charge • The potential due to charge q2 is • The equipotential surfaces for an isolated point charge are a family of spheres concentric with the charge E and V for an Infinite Sheet of Charge
• The equipotential lines are the dashed blue lines • The electric field lines are the brown lines • The equipotential lines are everywhere perpendicular to the field lines E and V for a Point Charge
• The equipotential lines are the dashed blue lines • The electric field lines are the brown lines • The equipotential lines are everywhere perpendicular to the field lines E and V for a Dipole
• The equipotential lines are the dashed blue lines • The electric field lines are the brown lines • The equipotential lines are everywhere perpendicular to the field lines Electric Potential for a Continuous Charge Distribution
• Consider a small charge element dq
– Treat it as a point charge • The potential at some point due to this charge element is V for a Continuous Charge Distribution, cont
• To find the total potential, you need to integrate to include the contributions from all the elements – This value for V uses the reference of V = 0 when P is infinitely far away from the charge distributions V for a Uniformly Charged Ring
• P is located on the perpendicular central axis of the uniformly charged ring
– The ring has a radius a and a total charge Q V Due to a Charged Conductor
• Consider two points on the surface of the charged conductor as shown • is always perpendicular to to the displacement • Therefore, =0 • Therefore, the potential difference between A and B is also zero V Due to a Charged Conductor, cont
• V is constant everywhere on the surface of a charged conductor in equilibrium V = 0 between any two points on the surface • The surface of any charged conductor in electrostatic equilibrium is an equipotential surface • Because the electric field is zero inside the conductor, we conclude that the electric potential is constant everywhere inside the conductor and equal to the value at the surface E Compared to V
• The electric potential is a function of r • The electric field is a function of r2 • The effect of a charge on the space surrounding it
– The charge sets up a vector electric field which is related to the force – The charge sets up a scalar potential which is related to the energy Definition of Capacitance
• Capacitors are devices that store electric charge • The capacitance, C, of a capacitor is defined as the ratio of the magnitude of the charge on either conductor to the potential difference between the conductors • The SI unit of capacitance is a farad (F) Makeup of a Capacitor
• A capacitor consists of two conductors
– When the conductors are charged, they carry charges of equal magnitude and opposite directions • A potential difference exists between the conductors due to the charge Field Due to a Plane of Charge (Ch19)
• The total charge in the surface is A • Applying Gauss’ Law • Note, this does not depend on r • Therefore, the field is uniform everywhere Parallel Plate Capacitor
• Each plate is connected to a terminal of the battery • If the capacitor is initially uncharged, the battery establishes an electric field in the connecting wires •The charge density on the plates is Capacitance Parallel Plate Assumptions • The assumption that the electric field is uniform is valid in the central region, but not at the ends of the plates • If the separation between the plates is small compared with the length of the plates, the effect of the nonuniform field can be ignored Energy in a Capacitor – Overview
• Consider the circuit to be a system • Before the switch is closed, the energy is stored as chemical energy in the battery • When the switch is closed, the energy is transformed from chemical to electric potential energy Capacitance of a Cylindrical Capacitor
• From Gauss’ Law, the field between the cylinders is E = 2 ke / r • V = 2 ke ln (b/a) • The capacitance becomes Circuit Symbols
• A circuit diagram is a simplified representation of an actual circuit • Circuit symbols are used to represent the various elements • Lines are used to represent wires • The battery’s positive terminal is indicated by the longer line Capacitors in Parallel
• When capacitors are first connected in the circuit, electrons are transferred from the left plates through the battery to the right plate, leaving the left plate positively charged and the right plate negatively charged Q = Q1 + Q2 V1 = V2 = Vbatt C=C1+C2 Capacitors in Parallel, 2
• The flow of charges ceases when the voltage across the capacitors equals that of the battery • The capacitors reach their maximum charge when the flow of charge ceases • The total charge is equal to the sum of the charges on the capacitors
– Q = Q1 + Q2 • The potential difference across the capacitors is the same
– And each is equal to the voltage of the battery Capacitors in Parallel, 3
• The capacitors can be replaced with one capacitor with a capacitance of Ceq
– The equivalent capacitor must have exactly the same external effect on the circuit as the original capacitors Capacitors in Series
• When a battery is connected to the circuit, electrons are transferred from the left plate of C1 to the right plate of C2 through the battery Capacitors in Series, 2
• As this negative charge accumulates on the right plate of C2, an equivalent amount of negative charge is removed from the left plate of C2, leaving it with an excess positive charge • All of the right plates gain charges of –Q and all the left plates have charges of +Q Equivalent Capacitance, Example • The 1.0μF and 3.0μF are in parallel as are the 6.0μF and 2.0μF • These parallel combinations are in series with the capacitors next to them • The series combinations are in parallel and the final equivalent capacitance can be found Energy Stored in a Capacitor
• Assume the capacitor is being charged and, at some point, has a charge q on it • The work needed to transfer a charge from one plate to the other is • The total work required is Energy, cont
• The work done in charging the capacitor appears as electric potential energy U • This applies to a capacitor of any geometry • The energy stored increases as the charge increases and as the potential difference increases • In practice, there is a maximum voltage before discharge occurs between the plates Energy, final
• The energy can be considered to be stored in the electric field • For a parallel plate capacitor, the energy can be expressed in terms of the field as • U= ( oAd)E2 • It can also be expressed in terms of the energy density (energy per unit volume) E2 uE = o Capacitors with Dielectrics
• A dielectric is an insulating material that, when placed between the plates of a capacitor, increases the capacitance
– Dielectrics include rubber, plastic, or waxed paper • With a dielectric, C = Co
– The capacitance is multiplied by the factor when the dielectric completely fills the region between the plates – For a parallel plate capacitor, this becomes – C = o(A/d) Dielectrics – An Atomic View
• The molecules that make up the dielectric are modeled as dipoles • The molecules are randomly oriented in the absence of an electric field Dielectrics – An Atomic View, cont
• An external electric field is applied • This produces a torque on the molecules • The molecules partially align with the electric field Dielectrics – An Atomic View, final
• An external field can polarize the dielectric whether the molecules are polar or nonpolar • The charged edges of the dielectric act as a second pair of plates producing an induced electric field in the direction opposite the original electric field The result of polarization Table of Some Dielectric Values V for a Uniformly Charged Sphere
• A solid sphere of radius R and total charge Q • For r > R, • For r < R, V for a Uniformly Charged Sphere, Graph
• The curve for VD is for the potential inside the curve
– It is parabolic – It joins smoothly with the curve for VB • The curve for VB is for the potential outside the sphere Problem Solving Strategies – Electric Potentials
• Conceptualize
– Think about the charges or the charge distribution – Image the type of potential they would create
• This establishes a mental representation – Use any symmetry in the arrangement of the charges to help you visualize the potential Problem Solving Strategies – Electric Potentials
• Categorize
– Individual charges or a distribution? • Analyze
– – – – – Scalar, so no components Superposition principle is algebraic sum Signs are important Changes in potential are what is important The point where V = 0 is arbitrary
• But usually at a point infinitely far from the charges Problem Solving Strategies – Electric Potentials, cont
• Analyze, cont
– For a group of individual charges, use the superposition principle – For a continuous charge distribution, integrate over the entire distribution – If is known, the line integral of can be evaluated Problem Solving Strategies – Electric Potentials, final
• Finalize
– Check to see if your result is consistent with the mental representation – Be sure the result reflects any symmetry you noted – Image varying parameters to see if the mathematical result changes in a reasonable way Summary and Hints
• Be careful with the choice of units
– In SI, capacitance is in F, distance is in m and the potential differences in V – Electric fields can be in V/m or N/c • When two or more capacitors are connected in parallel, the potential differences across them are the same
– The charge on each capacitor is proportional to its capacitance – The capacitors add directly to give the equivalent capacitance Summary and Hints, cont
• When two or more capacitors are connected in series, they carry the same charge, but the potential differences across them are not the same
– The capacitances add as reciprocals and the equivalent capacitance is always less than the smallest individual capacitor Finding E From V
• Assume, to start, that E has only an x component • Similar statements would apply to the y and z components • Equipotential surfaces must always be perpendicular to the electric field lines passing through them Electric Field from Potential, General
• In general, the electric potential is a function of all three dimensions • Given V (x, y, z) you can find Ex, Ey and Ez as partial derivatives ...
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This note was uploaded on 02/03/2010 for the course NEUROSCI 101A taught by Professor Scheibell during the Winter '10 term at UCLA.
 Winter '10
 Scheibell

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