 I1I1   R R R I2I2 I3I3

Today… Current and Current Density Devices –Batteries –Resistors Read Fishbane Chapter 26 Remember: Quiz on Thursday and Friday. – Covers chapters , – Potential, Capacitors, Gravity (and everything before)

Devices and Circuits We are now finished with electrostatics: the study of fields and potentials produced by static charge distributions. Next topic: Devices and Circuits We have studied one device so far: the capacitor. For the next week we will investigate circuits composed of the following devices: Capacitors Batteries Resistors } and combinations in “DC” circuits, (Direct Current circuits)

When an electric field is applied: – a small average velocity, v e,is added to the random motion (an electric current) Electrons exists in conductors with a density, n e (n e approx m -3 ) and are constantly in random motion –In general only electrons move, the heavy nucleii remain fixed in the material lattice In the absence of electric fields there is no net motion of the charge, electrons bounce around like atoms in a gas Current is charge in motion EE + - NOTE that the current direction is defined as the direction of the field BUT the electrons move in the opposite direction

Current is charge in motion Current density, J, is given by J = q e n e v e unit of J is C/m 2 sec or A/m 2 (A ≡ Ampere) and 1A ≡ 1C/s Current, I, is J times cross sectional area, I = J S –for 10 Amp in 1mm x 1mm area, J =10 +7 A/m 2, – v e is about m/s –(Yes, the average velocity is only 1mm/s!) EE + - n e electrons/m 3 Velocity v e Area S

Devices: Batteries Batteries (Voltage sources, sources of emf): Purpose is to provide a constant potential difference and source of current between two points. Cannot calculate the potential difference from first principles... chemical  electrical energy conversion. Non-ideal batteries will be dealt with in terms of an "internal resistance". Positive terminal has the higher potential Current is defined as flowing from the positive to the negative terminal Inside the battery chemical processes return the charge from the negative to positive terminals emf is the term for the electrical potential provided by the battery + - V + - OR

Devices: Resistors Resistors: Resistors limit the current drawn in a circuit. Resistance is a natural property of almost all materials which opposes the motion of charge through the material Resistance can be calculated from knowledge of the geometry of the resistor AND the “resistivity” of the material out of which it is made (often “conductors”). UNIT: Ampere = A = C/sNote:

Ohm’s Law Set up this circuit Vary applied voltage V. Measure current I Ratio remains constant Resistance R V I I R V I slope = R

Resistance What is happening in the resistance? V I I R Voltage means Potential Difference -> E -field E -field -> constant force on electrons Constant acceleration -> very large and increasing currents This does not happen large increasing currents are not observed –what’s wrong with this picture??? Constant force on electrons -> constant acceleration Electrons undergo a lot of rapid and random scattering No constant acceleration (acceleration proportional to Voltage) Instead velocity of electrons is proportional to Voltage velocity proportional to current -> I=V/R Simple constant acceleration isn’t happening….

What gives rise to non-ballistic behavior? E -field in conductor (resistor) provided by a battery Charges are put in motion, but scatter in a very short time from things that get in the way –it’s crowded inside that metal –defects, lattice vibrations (phonons), etc Typical scattering time  = sec Charges ballistically accelerated for this time and then randomly scattered

What gives rise to non-ballistic behavior? Newton’s 2 nd Law says F=ma So the acceleration of the electron is eE/m Average velocity attained between scatters is given by v=at or v = eE  / m Current density is J = env so current is proportional to E which is proportional to Voltage OHM’s LAW J = (e 2 n  /m)E or J =  E  = conductivity Or

Resistance Resistance is defined to be the ratio of the applied voltage to the current passing through. Is this a good definition? i.e., does the resistance belong only to the resistor? Recall the case of capacitance: ( C = Q / V ) depended on the geometry, not on Q or V individually Does R depend on V or I ? It seems as though it should, at first glance... V I I R UNIT: OHM = 

Calculating Resistance To calculate R, must calculate current I which flows when voltage V is applied. Applying voltage V sets up an electric field in the resistor. What determines the current? Current is charge flowing past a point per unit time, which depends on the average velocity of the charges. Field gives rise to force on the charge carriers which reach a terminal velocity. Resistance calculation splits into two parts –Part depends on the “resistivity” ρ, a property of the material –Part depends on the geometry (length L and cross sectional area A ) V I I R

Resistivity Where E = electric field and j = current density in conductor. L A E j Resistivity is a property of bulk matter related to the resistance of a sample. The resistivity (  ) is defined as: For the case of a uniform material

Resistivity L A E j e.g., for a copper wire,  ~  -m, 1mm radius, 1 m long, then R .01  for glass,  ~  -m; for semiconductors  ~ 1  -m So YES, the property belongs to the material and we can calculate the resistance if we know the resistivity and the dimensions of the object where

Makes sense? Increase the length, flow of electrons impeded Increase the cross sectional area, flow facilitated The structure of this relation is identical to heat flow through materials … think of a window for an intuitive example L A E j How thick? How big? What’s it made of? or

Question 1 Two cylindrical resistors, R 1 and R 2, are made of identical material. R 2 has twice the length of R 1 but half the radius of R 1. –The resistors are then connected to a battery V as shown V I1I1 I2I2 –What is the relation between the currents I 1 and I 2 (a) I 1 < I 2 (b) I 1 = I 2 (c) I 1 > I 2

Question 1 1.a 2.b 3.c

Two cylindrical resistors, R 1 and R 2, are made of identical material. R 2 has twice the length of R 1 but half the radius of R 1. –The resistors are then connected to a battery V as shown Question 1 V I1I1 I2I2 –What is the relation between the currents I 1 and I 2 (a) I 1 < I 2 (b) I 1 = I 2 (c) I 1 > I 2 The resistivity of both resistors is the same (  ). Therefore the resistances are related as: The resistors have the same voltage across them; therefore

Question 2 A very thin metal wire patterned as shown is bonded to some structure. As the structure is deformed this stretches the wire (slightly). –When this happens, the resistance of the wire: (a) decreases (b) increases (c) stays the same

Question 2 1.a 2.b 3.c

Question 2 A very thin metal wire patterned as shown is bonded to some structure. As the structure is deformed this stretches the wire (slightly). –When this happens, the resistance of the wire: (a) decreases (b) increases (c) stays the same Because the wire is slightly longer, is increased. Because the volume of the wire is ~constant, increasing the length, decreases the area, which increases the resistance. By carefully measuring the change in resistance, the strain in the structure may be determined

Is Ohm’s Law a good law? Our derivation of Ohm’s law ignored the effects of temperature. –At higher temperatures the random motion of electrons is faster, –time between collisions gets smaller –Resistance gets bigger –Temperature coefficient of resistivity (  ) –Typical values for metals  4  10 -3

Is Ohm’s Law a good law? Our derivation of Ohm’s law ignored quantum mechanical effects Many materials, only conduct when sufficient voltage is applied to move electrons into a “conduction band” in the material Examples are semiconductor diodes which have very far from linear voltage versus current plots

Is Ohm’s Law a good law? Superconductivity 1957: Bardeen, Cooper, and Schrieffer (“BCS”) publish theoretical explanation, for which they get the Nobel prize in – It was Bardeen’s second Nobel prize (1956 – transistor) –Current can flow, even if E=0. –Current in superconducting rings can flow for years with no decrease! At low temperatures (cooled to liquid helium temperatures, 4.2K)the resistance of some metals  0, measured to be less than ρ conductor (i.e., ρ< Ωm)!

Is Ohm’s Law a good law? Superconductivity 1986: “High” temperature superconductors are discovered (Tc=77K) –Important because liquid nitrogen (77 K) is much cheaper than liquid helium –Highest critical temperature to date ~140K Today: Superconducting loops are used to produce “lossless” electromagnets (only need to cool them, not fight dissipation of current) for particle physics. [Fermilab accelerator, IL] The Future: Smaller motors, “lossless” power transmission lines, magnetic levitation trains, quantum computers??...

Is Ohm’s Law a good law? Answer NO Ohm’s Law is not a fundamental law of physics However it is a good approximation for metallic conductors at room temperature as used in electrical circuits

Summary Ohm’s Law states Ohm’s Law is not a physical law but an approximation which works well enough in normal conditions Read Chapter 27 for tomorrow Remember the Quiz on Thursday and Friday.