EX. Potential for uniformly charged thin ring Q dq r R dφ 0 V ( Z ) =? z kdq Q Q V =, dq = Rdϕ = dϕ Q r 2πR 2π 2π k Q 0 = d ϕ 0 r 2π kq 0 2π = 0 d ϕ 2π r kq 0 = r kq 0 = 2 2 R + z
EX. Potential for uniformly charged spherical shell Q V =? V E r R dq R r R r kdq 0 V =, r Q Q 2 Q dq = da = R sinθdθdϕ = sin θdθdϕ, 2 2 4πR 4πR 4π kq 2π π 0 2 kq 0 V = sin R d d ( r R) 0 θ θ ϕ = > 0 4π r r r < R E = 0 V V = E ds 0 f i r r = V = constant in shpere
p r A B C Example : Potential due to an electric dipole Consider the electric dipole shown in the figure. We will determine the electric potential V created at point P by the two charges of the dipole using superposition. Point P is at a distance r from the center O of the dipole. Line OP makes an angle θ with the dipole axis 1 q q q r r V = V( + ) + V( ) = = 4πε o r r 4πε r r ( ) ( + ) ( + ) ( ) o ( ) ( + ) (24-6)
We assume that r? d where d is the p r charge separation. From triangle ABC we have: r( ) r( + ) dcosθ Also: q dcosθ 1 pcosθ r r r V = 2 ( ) ( + ) 2 2 4πε o r 4πε o r where p= qd = the electric dipole moment. A C V = 1 pcosθ 2 4πε o r B (24-6)
Example : O A dq Potential created by a line of charge of length L and uniform linear charge density λ at point P. Consider the charge element dq = λdx at point A, a distance x from O. From triangle OAP we have: 2 2 = + Here is the distance OP r d x The potential dv created by dq at P is: d (24-8)
O A dq dv V = 1 dq 1 λdx = = 4πε o r 4πε o d + x λ 4πε dx 2 o L 0 2 2 2 ( ) 2 2 x d x = ln + + λ V = ln x+ d + x 4πε V d + x o λ = ln + 4πε o d dx + x ( ) 2 2 ( ) 2 2 L L x 2 2 L 0 + lnd (24-8)
Determining E from V : ρ ρ ρ ρ dw = F ds = qe ds dw ρ ρ = E ds = dv q ( In general, ρ E = E iˆ + E E x x y ˆj + E kˆ z dv V V V =, E y =, E z =, x y z ˆ ˆ ρ i + j + kˆ, del) x y z E = V. E ρ q ds ρ if ( V E ds, f V i E = = dv ds E ds Fundamental principle of field parallel to path Calculus.) ρ V ˆ V ˆ V ( ˆ) (ˆ ˆ j k i j kˆ E = i + + = + + ) V. x y z x y z q EX. Check field for single charge r k q = r V 0 dv dv k0q E = = = 2 ds dr r ρ E = k0q rˆ 2 r EX. Check filed for uniformly charged thin ring Q V known V V E x = = 0 E y = = 0 x y E z V = z = dv dz =...
(24-9) r F( ) r F ( + ) Induced dipole moment Many molecules such as H O have a permanent electric dipole moment. These are known as "polar" molecules. Others, such as O, N, etc the electric 2 2 2 dipole moment is zero. These are known as "nonpolar" molecules. One such molecule is shown in fig.a. The electric dipole moment p r is zero because the center of the positive charge coincides with the center of the negative charge. In fig.b we show what happens when an electric field E r is applied on a nonpolar molecule. The electric forces on the positive and nagative charges are equal in magnitude but opposite in direction.
Work and Electric Potential Assume a charge moves in an electric field without any change in its kinetic energy The work performed on the charge is W = ΔV = q ΔV
U with Multiple Charges, final If there are more than two charges, then find U for each pair of charges and add them For three charges: U qq qq qq 1 2 1 3 2 3 = ke + + r12 r13 r23 The result is independent of the order of the charges
Finding E From V Assume, to start, that E has only an x component dv Ex = dx Similar statements would apply to the y and z components Equipotential surfaces must always be perpendicular to the electric field lines passing through them
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 Field from Potential, General In general, the electric potential is a function of all three dimensions Given V (x, y, z) you can find E x, E y and E z as partial derivatives V V V Ex = Ey = Ez = x y z
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 dq dv = ke r
V for a Continuous Charge Distribution, cont. To find the total potential, you need to integrate to include the contributions from all the elements V = k e dq r 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 dq kq e = k e = r x + a 2 2
V for a Uniformly Charged Disk The ring has a radius a and surface charge density of σ ( ) 1 2 2 2 V = 2πk eσ x + a x
V for a Finite Line of Charge A rod of line l has a total charge of Q and a linear charge density of λ V 2 2 kq e l + l + a = ln l a
V for a Uniformly Charged Sphere A solid sphere of radius R and total charge Q Q For r > R, V = ke r For r < R, kq e VD VC = R r 3 2R ( 2 2)
Irregularly Shaped Objects, cont. The field lines are still perpendicular to the conducting surface at all points The equipotential surfaces are perpendicular to the field lines everywhere
Cavity in a Conductor Assume an irregularly shaped cavity is inside a conductor Assume no charges are inside the cavity The electric field inside the conductor must be zero
Cavity in a Conductor, cont The electric field inside does not depend on the charge distribution on the outside surface of the conductor For all paths between A and B, V B V d A = E s = B A A cavity surrounded by conducting walls is a field-free region as long as no charges are inside the cavity 0
Corona Discharge If the electric field near a conductor is sufficiently strong, electrons resulting from random ionizations of air molecules near the conductor accelerate away from their parent molecules These electrons can ionize additional molecules near the conductor
Corona Discharge, cont. This creates more free electrons The corona discharge is the glow that results from the recombination of these free electrons with the ionized air molecules The ionization and corona discharge are most likely to occur near very sharp points
Millikan Oil-Drop Experiment Experimental Set-Up
Millikan Oil-Drop Experiment Robert Millikan measured e, the magnitude of the elementary charge on the electron He also demonstrated the quantized nature of this charge Oil droplets pass through a small hole and are illuminated by a light
Active Figure 25.27 (SLIDESHOW MODE ONLY)
Oil-Drop Experiment, 2 With no electric field between the plates, the gravitational force and the drag force (viscous) act on the electron The drop reaches terminal velocity with F D = mg
Oil-Drop Experiment, 3 When an electric field is set up between the plates The upper plate has a higher potential The drop reaches a new terminal velocity when the electrical force equals the sum of the drag force and gravity
Oil-Drop Experiment, final The drop can be raised and allowed to fall numerous times by turning the electric field on and off After many experiments, Millikan determined: q = ne where n = 1, 2, 3, e = 1.60 x 10-19 C
Van de Graaff Generator Charge is delivered continuously to a high-potential electrode by means of a moving belt of insulating material The high-voltage electrode is a hollow metal dome mounted on an insulated column Large potentials can be developed by repeated trips of the belt Protons accelerated through such large potentials receive enough energy to initiate nuclear reactions
Electrostatic Precipitator An application of electrical discharge in gases is the electrostatic precipitator It removes particulate matter from combustible gases The air to be cleaned enters the duct and moves near the wire As the electrons and negative ions created by the discharge are accelerated toward the outer wall by the electric field, the dirt particles become charged Most of the dirt particles are negatively charged and are drawn to the walls by the electric field
(24-9) r F( ) r F ( + ) Induced dipole moment Many molecules such as H O have a permanent electric dipole moment. These are known as "polar" molecules. Others, such as O, N, etc the electric 2 2 2 dipole moment is zero. These are known as "nonpolar" molecules. One such molecule is shown in fig.a. The electric dipole moment p r is zero because the center of the positive charge coincides with the center of the negative charge. In fig.b we show what happens when an electric field E r is applied on a nonpolar molecule. The electric forces on the positive and nagative charges are equal in magnitude but opposite in direction.
Capacitance V + V - A system of two isolated conductors separated by an insulator (this can be vacuum or air) one with a charge +q and the other q is known as a capacitor. The symbol used to indicate a capacitor is two parallel lines. We refer to the conductors as plates. We refer to the charge of the capacitor as the absolute value of the charge on either plate C = q V (25-2)
Makeup of a Capacitor A capacitor consists of two conductors These conductors are called plates When the conductor is charged, the plates carry charges of equal magnitude and opposite directions A potential difference exists between the plates due to the charge
More About Capacitance Capacitance will always be a positive quantity The capacitance of a given capacitor is constant The capacitance is a measure of the capacitor s ability to store charge The farad is a large unit, typically you will see microfarads (μf) and picofarads (pf)
Definition of Capacitance 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 Q C = ΔV The SI unit of capacitance is the farad (F)
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
Parallel Plate Capacitor, cont This field applies a force on electrons in the wire just outside of the plates The force causes the electrons to move onto the negative plate This continues until equilibrium is achieved The plate, the wire and the terminal are all at the same potential At this point, there is no field present in the wire and the movement of the electrons ceases
Parallel Plate Capacitor, final The plate is now negatively charged A similar process occurs at the other plate, electrons moving away from the plate and leaving it positively charged In its final configuration, the potential difference across the capacitor plates is the same as that between the terminals of the battery
Capacitance Isolated Sphere Assume a spherical charged conductor Assume V = 0 at infinity C Q Q R = = = = ΔV k Q/ R k e 4πε R Note, this is independent of the charge and the potential difference e o
Energy stored in capacitor q + q _ q q dwapp = Vdq = dq C Q q W = dq 0 c 1 2 Q C +Q Q A d ε 2 2 2 app = = PEE = U E = = 1 2 C V C 1 2 CV C 2 A = ε d 1 Without battery:(q fixed) 2 A Q Q 1, fixed = =, =. d V C o 1 d A : C ε Ue 2 2, fixed = A Q Q,. d = V = C o 1 A d : C ε Ue 2 2