Chapter 21-part1 Current and Resistance

1 Electric Current Whenever electric charges move, an electric current is said to exist Whenever electric charges move, an electric current is said to exist The current is the rate at which the charge flows through a certain cross- section The current is the rate at which the charge flows through a certain cross- section For the current definition, we look at the charges flowing perpendicularly to a surface of area A For the current definition, we look at the charges flowing perpendicularly to a surface of area A

Definition of the current: Charge in motion through an area A. The time rate of the charge flow through A defines the current (=charges per time): I=  Q/  t Units: C/s=As/s=A SI unit of the current: Ampere + -

Electric Current, cont The direction of current flow is the direction positive charge would flow The direction of current flow is the direction positive charge would flow This is known as conventional (technical) current flow, i.e., from plus (+) to minus (-) This is known as conventional (technical) current flow, i.e., from plus (+) to minus (-) However, in a common conductor, such as copper, the current is due to the motion of the negatively charged electrons However, in a common conductor, such as copper, the current is due to the motion of the negatively charged electrons It is common to refer to a moving charge as a mobile charge carrier It is common to refer to a moving charge as a mobile charge carrier A charge carrier can be positive or negative A charge carrier can be positive or negative

2 Current and Drift Speed Charged particles move through a conductor of cross- sectional area A Charged particles move through a conductor of cross- sectional area A n is the number of charge carriers per unit volume V (=“concentration”) n is the number of charge carriers per unit volume V (=“concentration”) nA  x=nV is the total number of charge carriers in V nA  x=nV is the total number of charge carriers in V

Current and Drift Speed, cont The total charge is the number of carriers times the charge per carrier, q (elementary charge) The total charge is the number of carriers times the charge per carrier, q (elementary charge) ΔQ = (nAΔx)q [unit: (1/m 3 )(m 2 m)As=C] ΔQ = (nAΔx)q [unit: (1/m 3 )(m 2 m)As=C] The drift speed, v d, is the speed at which the carriers move The drift speed, v d, is the speed at which the carriers move v d = Δx/Δt v d = Δx/Δt Rewritten: ΔQ = (nAv d Δt)q Rewritten: ΔQ = (nAv d Δt)q Finally, current, I = ΔQ/Δt = nqv d A Finally, current, I = ΔQ/Δt = nqv d A ΔxΔxΔxΔx

Current and Drift Speed, final If the conductor is isolated, the electrons undergo (thermal) random motion If the conductor is isolated, the electrons undergo (thermal) random motion When an electric field is set up in the conductor, it creates an electric force on the electrons and hence a current When an electric field is set up in the conductor, it creates an electric force on the electrons and hence a current

Charge Carrier Motion in a Conductor The electric field force F imposes a drift on an electron’s random motion (10 6 m/s) in a conducting material. Without field the electron moves from P 1 to P 2. With an applied field the electron ends up at P 2 ’; i.e., a distance v d  t from P 2, where v d is the drift velocity (typically m/s). The electric field force F imposes a drift on an electron’s random motion (10 6 m/s) in a conducting material. Without field the electron moves from P 1 to P 2. With an applied field the electron ends up at P 2 ’; i.e., a distance v d  t from P 2, where v d is the drift velocity (typically m/s).

Does the direction of the current depend on the sign of the charge? No! ( a) Positive charges moving in the same direction of the field produce the same positive current as (b) negative charges moving in the direction opposite to the field. ( a) Positive charges moving in the same direction of the field produce the same positive current as (b) negative charges moving in the direction opposite to the field. E E vdvd vdvd qvdqvd (-q)(-v d ) = qv d

Current density: The current per unit cross-section is called the current density J: The current per unit cross-section is called the current density J: J=I/A= nqv d A/A=nqv d In general, a conductor may contain several different kinds of charged particles, concentrations, and drift velocities. Therefore, we can define a vector current density: In general, a conductor may contain several different kinds of charged particles, concentrations, and drift velocities. Therefore, we can define a vector current density: J=n 1 q 1 v d1 +n 2 q 2 v d2 +… J=n 1 q 1 v d1 +n 2 q 2 v d2 +… Since, the product qv d is for positive and negative charges in the direction of E, the vector current density J always points in the direction of the field E. Since, the product qv d is for positive and negative charges in the direction of E, the vector current density J always points in the direction of the field E.

Example: An 18-gauge copper wire (diameter 1.02 mm) carries a constant current of 1.67 A to a 200 W lamp. The density of free electrons is 8.5  per cubic meter. Find the magnitudes of (a) the current density and (b) the drift velocity. An 18-gauge copper wire (diameter 1.02 mm) carries a constant current of 1.67 A to a 200 W lamp. The density of free electrons is 8.5  per cubic meter. Find the magnitudes of (a) the current density and (b) the drift velocity.

Solution: (a) A=d 2  /4=( m) 2  /4=8.2  m 2 J=I/A=1.67 A/(8.2  m 2 )=2.0  10 6 A/m 2 (b) From J=I/A=nqv d, it follows: v d =1.5  m/s=0.15 mm/s

3 Electrons in a Circuit The drift speed is much smaller than the average speed between collisions The drift speed is much smaller than the average speed between collisions When a circuit is completed, the electric field travels with a speed close to the speed of light When a circuit is completed, the electric field travels with a speed close to the speed of light Although the drift speed is on the order of m/s the effect of the electric field is felt on the order of 10 8 m/s Although the drift speed is on the order of m/s the effect of the electric field is felt on the order of 10 8 m/s

Meters in a Circuit – Ammeter An ammeter is used to measure current In line with the bulb, all the charge passing through the bulb also must pass through the meter (in series!)

Meters in a Circuit - Voltmeter A voltmeter is used to measure voltage (potential difference) Connects to the two ends of the bulb (parallel)

QUICK QUIZ Look at the four “circuits” shown below and select those that will light the bulb.

4 Resistance and Ohm’s law In a homogeneous conductor, the current density is uniform over any cross section, and the electric field is constant along the length. In a homogeneous conductor, the current density is uniform over any cross section, and the electric field is constant along the length. a b V=V a -V b =EL

Resistance The ratio of the potential drop to the current is called resistance of the segment: The ratio of the potential drop to the current is called resistance of the segment: Unit: V/A=  ohm 

Resistance, cont Units of resistance are ohms (Ω) Units of resistance are ohms (Ω) 1 Ω = 1 V / A 1 Ω = 1 V / A Resistance in a circuit arises due to collisions between the electrons carrying the current with the fixed atoms inside the conductor Resistance in a circuit arises due to collisions between the electrons carrying the current with the fixed atoms inside the conductor

Ohm’s Law V  I  V=const.  I  V=RI V  I  V=const.  I  V=RI Ohm’s Law is an empirical relationship that is valid only for certain materials Ohm’s Law is an empirical relationship that is valid only for certain materials Materials that obey Ohm’s Law are said to be ohmic Materials that obey Ohm’s Law are said to be ohmic I=V/R I=V/R R , I  0, open circuit; R  0, I , short circuit R , I  0, open circuit; R  0, I , short circuit

Ohm’s Law, final Plots of V versus I for (a) ohmic and (b) nonohmic materials. The resistance R=V/I is independent of I for ohmic materials, as is indicated by the constant slope of the line in (a). Plots of V versus I for (a) ohmic and (b) nonohmic materials. The resistance R=V/I is independent of I for ohmic materials, as is indicated by the constant slope of the line in (a). Ohmic Nonohmic

5 Resistivity Expected: R  L/A Expected: R  L/A The resistance of an ohmic conductor is proportional to its length, L, and inversely proportional to its cross-sectional area, A The resistance of an ohmic conductor is proportional to its length, L, and inversely proportional to its cross-sectional area, A ρ (“rho”) in  m is the constant of proportionality and is called the resistivity of the material ρ (“rho”) in  m is the constant of proportionality and is called the resistivity of the material

Example Determine the required length of nichrome (  =10 -6  m) with a radius of 0.65 mm in order to obtain R=2.0 . Determine the required length of nichrome (  =10 -6  m) with a radius of 0.65 mm in order to obtain R=2.0 . R=  L/A  L=RA/  R=  L/A  L=RA/ 

The resistivity depends on the material and the temperature The resistivity depends on the material and the temperature

6 Temperature Variation of Resistivity For most metals, resistivity increases with increasing temperature For most metals, resistivity increases with increasing temperature With a higher temperature, the metal’s constituent atoms vibrate with increasing amplitude With a higher temperature, the metal’s constituent atoms vibrate with increasing amplitude The electrons find it more difficult to pass the atoms (more scattering!) The electrons find it more difficult to pass the atoms (more scattering!)

Temperature Variation of Resistivity, cont For most metals, resistivity increases approximately linearly with temperature over a limited temperature range For most metals, resistivity increases approximately linearly with temperature over a limited temperature range ρ o is the resistivity at some reference temperature T o ρ o is the resistivity at some reference temperature T o T o is usually taken to be 20° C T o is usually taken to be 20° C  is the temperature coefficient of resistivity [unit: 1/(  C)]  is the temperature coefficient of resistivity [unit: 1/(  C)]

Temperature Variation of Resistance Since the resistance of a conductor with uniform cross sectional area is proportional to the resistivity, the temperature variation of resistance can be written Since the resistance of a conductor with uniform cross sectional area is proportional to the resistivity, the temperature variation of resistance can be written

Example The material of the wire has a resistivity of  0 =6.8   m at T 0 =320  C, a temperature coefficient of  =2.0  (1/  C) and L=1.1 m. Determine the resistance of the heater wire at an operating temperature of 420  C. The material of the wire has a resistivity of  0 =6.8   m at T 0 =320  C, a temperature coefficient of  =2.0  (1/  C) and L=1.1 m. Determine the resistance of the heater wire at an operating temperature of 420  C.

Solution  =  0 [1+  0 )]  =  0 [1+  0 )]  =[6.8   m][1+(2.0  (  C) -1 )   =[6.8   m][1+(2.0  (  C) -1 )   (420  C-320  C)]=8.2   m  (420  C-320  C)]=8.2   m R=  L/A R=  L/A R=(8.2   m)(1.1 m)/(3.1  m 2 ) R=(8.2   m)(1.1 m)/(3.1  m 2 ) R=29  R=29 

7 Superconductors A class of materials and compounds whose resistances fall to virtually zero below a certain temperature, T C A class of materials and compounds whose resistances fall to virtually zero below a certain temperature, T C T C is called the critical temperature (in the graph 4.1 K) T C is called the critical temperature (in the graph 4.1 K) “normal”

Superconductors, cont The value of T C is sensitive to The value of T C is sensitive to Chemical composition Chemical composition Pressure Pressure Crystalline structure Crystalline structure Once a current is set up in a superconductor, it persists without any applied voltage Once a current is set up in a superconductor, it persists without any applied voltage Since R = 0 Since R = 0

Superconductor Timeline Superconductivity discovered by H. Kamerlingh Onnes Superconductivity discovered by H. Kamerlingh Onnes High-temperature superconductivity discovered by Bednorz and Müller High-temperature superconductivity discovered by Bednorz and Müller Superconductivity near 30 K Superconductivity near 30 K Superconductivity at 92 K and 105 K Superconductivity at 92 K and 105 K Current Current More materials and more applications More materials and more applications

T c values for different materials; note the high T c values for the oxides. T c values for different materials; note the high T c values for the oxides.

It’s magic! It’s magic!

8 Electrical Energy and Power In a circuit, as a charge moves through the battery, the electrical potential energy of the system is increased by ΔQΔV [AsV=Ws=J] In a circuit, as a charge moves through the battery, the electrical potential energy of the system is increased by ΔQΔV [AsV=Ws=J] The chemical potential energy of the battery decreases by the same amount The chemical potential energy of the battery decreases by the same amount As the charge moves through a resistor, it loses this potential energy during collisions with atoms in the resistor As the charge moves through a resistor, it loses this potential energy during collisions with atoms in the resistor The temperature of the resistor will increase The temperature of the resistor will increase

Electrical Energy and Power, cont The rate of the energy transfer is power (P): V Units: (C/s)(J/C) =J/s=W 1J=1Ws=1Nm W=AV

Electrical Energy and Power, cont From Ohm’s Law, alternate forms of power are (use V=IR and I=V/R) From Ohm’s Law, alternate forms of power are (use V=IR and I=V/R) Joule heat (I 2 R losses)

Electrical Energy and Power, final The SI unit of power is Watt (W) The SI unit of power is Watt (W) I must be in Amperes, R in Ohms and V in Volts I must be in Amperes, R in Ohms and V in Volts The unit of energy used by electric companies is the kilowatt-hour The unit of energy used by electric companies is the kilowatt-hour This is defined in terms of the unit of power and the amount of time it is supplied This is defined in terms of the unit of power and the amount of time it is supplied 1 kWh =(10 3 W)(3600 s)= 3.60 x 10 6 J 1 kWh =(10 3 W)(3600 s)= 3.60 x 10 6 J

9 Electrical Activity in the Heart Every action involving the body’s muscles is initiated by electrical activity Every action involving the body’s muscles is initiated by electrical activity Voltage pulses cause the heart to beat Voltage pulses cause the heart to beat These voltage pulses (  1 mV) are large enough to be detected by equipment attached to the skin These voltage pulses (  1 mV) are large enough to be detected by equipment attached to the skin Heart beat Initiation

Electrocardiogram (EKG) A normal EKG A normal EKG P occurs just before the atria begin to contract P occurs just before the atria begin to contract The QRS pulse occurs in the ventricles just before they contract The QRS pulse occurs in the ventricles just before they contract The T pulse occurs when the cells in the ventricles begin to recover The T pulse occurs when the cells in the ventricles begin to recover

Abnormal EKG, 1 The QRS portion is wider than normal The QRS portion is wider than normal This indicates the possibility of an enlarged heart This indicates the possibility of an enlarged heart

Abnormal EKG, 2 There is no constant relationship between P and QRS pulse This suggests a blockage in the electrical conduction path between the SA and the AV nodes This leads to inefficient heart pumping

Abnormal EKG, 3 No P pulse and an irregular spacing between the QRS pulses Symptomatic of irregular atrial contraction, called fibrillation The atrial and ventricular contraction are irregular

Implanted Cardioverter Defibrillator (ICD) Devices that can monitor, record and logically process heart signals Devices that can monitor, record and logically process heart signals Then supply different corrective signals to hearts that are not beating correctly Then supply different corrective signals to hearts that are not beating correctly Monitor lead Dual chamber ICD