Chapter 30: Potential and Field (aka Chapter 29 The Sequel )
Electric Field and Electric Potential: Two Sides of the Same Coin A set of charges ( source charges ) alters the space around them. This alteration of space exerts a force on other charged particles ( test charges ) The alteration of space can be represented either by the the electric field E or by the electric potential V E is a vector field V is a scalar field Both quantities give us the exact same information How can that be???
But doesn t E have more information than V? E has three components at each point in space!" E = Ex (r " )i # + E y (r " ) # j + E z (r " )k # V has only a single component at each point in space V = V(r! ) Force is conservative! For time independent behavior, V completely determines E For time varying systems (eg. electrical generators, radio transmitters, electromagnetic waves such as light or radio), V does not fully determine E. This case will come later in the course
E is to V what F is to U Getting the potential energy from force: f!u = "W (i # f ) = " $ F!" ids " i Getting the force from the potential F x = " du (for 1D) dx!" & %U F = " %x i# + %U %y j# + %U ' ( %z k# ) * + (for 2D or 3D)
Finding the Potential from the Electric Field Start with: V = U / q and F!" = qe!" The potential difference between two points in space is We can think of an integral as an area under a curve. Thus a graphical interpretation of the equation above is
Getting V from E: One dimension E x = E 0 x! "V = V(x) # V(0) = # E x (x')dx' = # 1 2 E 0x 2 Let V(0)=0 since only the change in V has any physical meaning $ V(x) == # E x (x')dx' = # 1 2 E 0x 2 0 x $ 0 x If E x (x) = E 0 x n then V(x) = V(0)! 1 (n + 1) E 0 xn+1 valid for all n positive and negative except for n=-1
Potential for a Point Charge: What to do in 3D Finding V from the electric field is easy in 1D because the integral is very simple. How to do the line integral in 3D?!" 1 q E = 4!" 0 r r# 2 For a point particle, E points radially away. Therefore integration along a surface of constant distance from the charge (sphere), gives 0. Only contribution to line integral is for integration that is in the direction of increasing distance from the charge. Therefore to get V we integrate along r!. Any other line integral will simply have additional terms that give 0. #!V = V(r) " V(#) = " $ E!" i("r # )dr ' = q # 1 r 4%& 0 $ = " q 1 r r 2 4%& 0 r ' r V(r) = V(#) + q 4%& 0 r We want V(!)=0 since it makes no sense for a charge to produce a finite potential at the other end of the universe. We will always let V(!)=0 when E!" "1 / r n #
Charge Separation Leads to Potential Difference Two spatially separated regions of opposite charge lead to a potential difference negative!v = V positive " V negative = " E!" ids " # positive This is how a battery works: Chemical reactions separate positive and negative ions leading to potential difference between terminals.
A Simple Picture of How a Battery Works 1. A battery has two terminals (+ and -). Each is made of a different metal or metal oxide (like rust) 2. Electrodes are in contact with an electrolyte solution such as salt in water or an acid in water. 3. The different metals of the terminals react differently with the salt or acid of the electrolyte solution 5. The different reactions allow positive ions to be produced at one terminal and negative ions at the other terminal. 6. Reaction only proceeds if the two terminals are connected by a conductor.
Batteries do Work on the Ions Chemical reactions in the battery release energy This energy does work that separates the ions and creates a potential difference between the terminals. Chemical Energy Mechanical Work Electrical Potential Energy Eventually, all the salt or acid in solution has reacted with the terminals--> Battery dies. The amount of work done per unit of charge is known as emf.
The Electric Battery: History Luigi Galvani (Italy, 1780 s) studied the effect of static electricity on the contraction of leg muscles in frogs, and found that the same effect could be produced by inserting two dissimilar metals into the muscle. Alessandro Volta (Italy,1800) invented the electric battery and demonstrated a flow of electric charge. Volta s original battery consisted of alternate layers of zinc and silver and a salt solution. A simple electric cell is the basis of the common 1.5 Volt dry cell flashlight battery.
Common dry cell. (AA,AAA, C or D cell) Common Dry cell The electrolyte (acid) reacts with the zinc electrode dissolving part of it. Each zinc atoms enters solution as a positive ion, leaving two electrons behind. The zinc electrode is left with a net negative charge. The positively charged electrolyte (Zn ions) pulls electrons off the carbon electrode, leaving it with a net positive charge. When connected to a circuit, electrons at the negative zinc electrode travel through the circuit to the carbon electrode (positive). The net result is that the carbon electrode is left with a net positive charge and the zinc electrode a net negative charge, creating a potential difference between them of 1.5 V.
Car Battery (Lead-Acid) (Reaction at Pb electrode) (Reaction at PbO 2 electrode) This leads to an energy release per electron (emf) of 2.2 V (ideal case). Real batteries have losses so it is closer to 2V. Car batteries have 6 such individual batteries in series 6*2V=12V
Lead-Acid Car Battery (2) 1. If there is no wire connecting the terminals, electrons and H+ ions build up at the Pb (-) terminal. H+ ions can not migrate freely to PbO 2 terminal and second reaction does not occur. 2. Electrons from the Pb (-) electrode move through a circuit connecting the electrodes to the PbO 2 positive electrode where they undergo the second reaction.
Finding the Electric Field from the Potential s!v = V(s i ) " V(s f ) = " E!" ids " f s f # = "# E s ds For small!s = s i " s f,!v $ "E s!s!v E s = " lim!s%0!s = " dv ds In terms of the potential, the component of the electric field in the s-direction is s i s i Generalize to the three Cartesian axes...
The Gradient of a Scalar Field!(x, y) = sin(x)sin(y) "!"! = #! #x i# + #! #x j# = cos(x)sin(y)i # + sin(x)cos(y) j # The gradient!!" " always points "uphill" The electric field would be E!" = #!!" " and points "downhill"
Using V to Determine E!"!" $ #V E =!" V =! #x i# + #V #y j# + #V & % #z k# ' ) ( You must understand this formula!!!
V(x, y, z) = ( 100V / m 4 ) x 2 yz -First treat y and z as constants... Example 1!V!x =! "( 100V / m 4 )( yz) x 2 $!x # % -Second treat x and z as constants...!v!y =! " 100V / m 4!y # ( )( x 2 z) y -Lastly treat x and y as constants...!v!z =! "( 100V / m 4 )( x 2 y)z$!z # % Assemble final answer:! E = &!V î &!V ĵ &!V ˆk!x!y!z! E = & 100V / m 4 ( ) yz $ % = ( 100V / m4 )( yz)!!x x2 = ( 100V / m 4 )( yz)2x = ( 100V / m4 )( x 2 z)!!y = ( 100V / m4 )( x 2 y)!!z ( ) ĵ + ( x2 y) ˆk (( )2xî + x2 z ) y = ( 100V / m4 )( x 2 z) z = ( 100V / m4 )( x 2 y)
V(x, y,z) = ( 100V / m 2 )( x 2 + y 2 + z 2 ) -First treat y and z as constants...!v!x =! "( 100V / m 2 )( x 2 + y 2 + z 2 ) $!x # % -Second treat x and z as constants...!v!y =! " 100V / m 4!y # ( )( x 2 + y 2 + z 2 ) -Lastly treat x and y as constants...!v!z =! " 100V / m 4!z # Assemble final answer:! E = &!V î &!V ĵ &!V!x!y!z! E = & 100V / m 2 ( )( x 2 + y 2 + z 2 ) ( ) 2xî ˆk ( + 2yĵ + 2z ˆk ) Example 2 $ % $ % = ( 100V / m2 )!!x x2 = ( 100V / m 2 )2x = ( 100V / m4 )!!y y2 = ( 100V / m 2 )2y = ( 100V / m4 )!!z z2 = ( 100V / m 2 )2z
Geometric Relation Between Potential and Electric Field Along an equipotential surface the potential is unchanged (V=const.) Case 1:!s! 1 is tangent to equipotential E s1 " #!V!s 1 = # V # # V #!s 1 = 0 Case 2:!s! 2 is perpendicular to the equipotential E s2 " #!V!s 2 = # V + # V #!s 2 < 0 The electric field is always perpendicular to the equipotential surfaces and points towards decreasing potential ( downhill ) V + >V -
Kirchhoff s Loop Law (VERY IMPORTANT) For any path that starts and ends at the same point This statement is known as Kirchhoff s loop law. It is VERY IMPORTANT for analyzing how electric circuits work. (It works for gravity too!!!!!)
How to Derive Kirchoff s Law V(b) The electric force is conservative! The potential energy difference is independent of the path taken. Therefore V(a)! V(b) =! E!" ids " 1 V(a)! V(b) =! E!" ids " 2 b " a b " a 0 = V(a)! V(a) = (V(a)! V(b))! (V(a)! V(b)) b 0 =! E!" ids " b # & " 1 % ( $ a '!! E!" ids " b # & " 2 % ( $ a ' =! E!" ids " a # 1 + E!" i(!ds " & " 2 % " )( $ a b ' 0 = #" E!" ids " This closed line integral where the integration makes a closed loop and ends at the starting point. It can be rewritten in terms of the voltage drops along: 0 = #" E!" ids " = E!" ids " x 1" x 2" + E!" ids " x 3 + E!" ids " a x 1 x n+1 " +... + " E!" ids " b + " E!" ids " 0 = )V a* x1 + )V x1 * x 2 + )V x2 * x 3 +... + )V xn * x n+1 + )V xn+1 *b x 2 x n x n+1 V(a) s 1 s 2
Conductors in Electrostatic Equilibrium: The Potential!" E = 0 inside a conductor in equilibrium (if not, currents would flow and it would not be equilibrium) b!v=-" E!" ids " b = " 0 ds = 0 a a All points are at the same potential. Surface is thus an equipotential. Electric field always perpendicular to equipotentials.
Capacitors and Capacitance Consider 2 conductors a certain distance apart with +Q and -Q charges The surfaces are equipotentials b!" E! Q, "Vc = V(a) # V(b) = # $ E!" ids " = Q % (constant depending on size, shape, and distance) a "V c = Q % 1 C C is a purely geometric quantity
Capacitance and Capacitors The ratio of the charge Q to the potential difference ΔV C is called the capacitance C. Capacitance is a purely geometric property of two electrodes TheSI unit of capacitance is the farad: 1 farad = 1 F = 1 C/V. Capacitance for parallel plate capacitor: A=area of plate
Capacitors are Everywhere: Not Just Parallel Plates. Two Examples.
Nested Spherical Shells Two concentric spherical shells First has radius R 1 and charge Q 1 =+Q Second has radius R 2 and charge Q 2 =-Q The field in between spheres is due to the inner sphere alone (Gauss s Law) R 2!V c = "# E!" i ˆrdr = " 1 R 1 Q 4$% 0 # dr = 1 & Q " Q ) r 2 4$% 0 ' ( R 1 R 2 * + = Q & R " R ) 2 1 ' ( 4$% 0 R 1 R 2 * + R 1 R 2 C = Q /!V c = 4$% 0 R 1 R 2 R 2 " R 1 We can rewrite this as a ratio of surface area to separation R 2 " R 1 = d # R 2, R 1 R C = 4$% 1 R 2 R 0 = 4$% 1 (d + R 1 ) 0 R 2 " R 1 d, % 0 4$ R 1 2 d = % 0 A sphere d Same form as for ideal parallel plate capacitor
Cylindrical Capacitor (Coaxial Cable for TV) a < r < b : E!" =! 2"# 0 r ˆr (from Gauss s Law)!= charge per unit length b "V c =! 1 2#$ 0 % r dr =! lnb & lna 2#$ 0 C =!L / "V c = 2#$ 0L ln(b / a) a ( ) =! 2#$ 0 ln( b a )
Charging a Capacitor (Reminder from Ch. 29) Connect capacitor to battery. Initially current flows that charges capacitor plates When capacitor voltage is same as EMF of battery, current stops flowing. Circuit Diagrams: Battery: Capacitor: = =
EXAMPLE 30.6 Charging a capacitor Typical capacitors have capacitances from 1µ F = 10!6 F to 1pF = 10!12 F Although this is a commonplace capacitance for circuits, why is A so large?
In Series: Batteries in Series and Parallel In parallel:!v =!V 1 +!V 2 Voltage difference is still!v, unchanged. (Delivers more current longer. eg. larger computer batteries last longer)
Capacitors in Parallel Both capacitors are connected to the battery: Both have the same potential as the battery.
Equivalent Capacitance (Parallel) Two capacitors in parallel are equivalent to one capacitor with same total charge.
Capacitors in Series The two capacitors are no longer directly in contact with battery: Different potentials now! Kirchoff's Law:!V battery =!V 1 +!V 2
Equivalent Capacitance (Series) The charge on both capacitors must be the same! They are equivalent to one capacitor with potential!v
Combinations of Capacitors If capacitors C 1, C 2, C 3, are in parallel, their equivalent capacitance is (Parallel capacitors have the same electric potential) If capacitors C 1, C 2, C 3, are in series, their equivalent capacitance is (Series capacitors have the same amount of charge.)
The Energy Stored in a Capacitor Capacitors are important elements in electric circuits because of their ability to store energy. In terms of the capacitor s potential difference, the potential energy stored in a capacitor is U C = Q2 2C = 1 2 C!V 2 Charging a capacitor increases its potential energy. The potential energy comes from the battery
Capacitors can be charged very slowly by a battery and then release a large amount of energy very quickly (high power) Examples: camera flash, defibrillator
EXAMPLE 30.9 Storing energy in a capacitor
EXAMPLE 30.9 Storing energy in a capacitor
The Energy in the Electric Field U C = 1 2 C!V 2 # = " 0 2 E 2 $ % & ' ( ) ( Ad) = ( energy per unit volume) ) (volume) The capacitor s energy is stored in the electric field! The energy density of an electric field, such as the one inside a capacitor, is The energy density has units J/m 3.
Dielectrics A dielectric is an insulating material. It does not conduct electricity but it can be polarized by an electric field. What happens if we place a dielectric inside a capacitor? Charge on the plates remains fixed. What happens to the voltage? insulator!v =!V 0 /" C = Q /!V = " (Q 0 /!V 0 ) = "C 0 Capacitance changes by an amount "
Induced Dipoles of Dielectric Change E!" Etotal = E!" plates + E!" induced E total = E 0! E induced = E 0! "E 0 = (1! ")E 0 E total = #!1 E 0 # > 1
Dielectric Dielectrics weaken the total electric field inside the capacitor. This implies: For fixed charge on the plates, the voltage across the capacitor decreases. For fixed voltage across the capacitor, the amount of charge on the plates increases C = Q /!V = "Q 0 /!V # Q = "Q 0
Dielectrics The dielectric constant,", like density or specific heat, is a property of a material. Easily polarized materials have larger dielectric constants than materials not easily polarized. Vacuum has κ = 1 exactly. Adding a dielectric increases the capacitance of a capacitor by a factor of "