1 1 W06D1 Current, Current Density, Resistance and Ohm’s Law, Magnetic Field, Magnetic Force Today’s Reading Assignment: Current, Current Density, and Resistance and Ohm’s Law, Magnetic Fields and Forces Course Notes: Sections 6.1-6.5, 8.1-8.3, 8.5 Announcements Week 6 Problem Solving and Math Review Tuesday from 9-11 pm in 26-152 PS 5 due Week 6 Tuesday at 9 pm in boxes outside 32-082 or 26-152 W06D2 Reading Assignment Course Notes: Magnetic Forces, Currents & Dipoles; Sections 8.3, 9.1-9.2 Exam 2 Thursday March 20 7:30 - 9:30 pm 2 3 Outline Current and Current Density Resistance and Ohm’s Law Magnetic Field Magnetic Forces2 4 Current: Flow Of Charge Units of Current: Coulomb/second = Ampere Average current Iav: Charge flowing across area A in time Iav=ΔQΔtInstantaneous current: differential limit of Iav dQIdt= ΔQ Δt5 How Big is an Ampere? • Household Electronics • Battery Powered • Household Service • Lightning Bolt • To hurt you • To throw you • To kill you • Fuse/Circuit Breaker ~1 A ~100 mA (1-10 A-Hr) 100 A 10 to 100 kA 40 (5) mA DC(AC) 60 (15) mA DC(AC) 0.5 (0.1) A DC(AC) 15-30 A 6 Direction of the Current Direction of current is direction of flow of pos. charge or, opposite direction of flow of negative charge3 7 Why Does Current Flow? If an electric field is set up in a conductor, charge will move (making a current in direction of E)Note that when current is flowing, the conductor is not an equipotential surface (and Einside ≠ 0)! 8 Microscopic Picture Drift velocity is the average velocity forced by applied electric field in the presence of collisions. Magnitude is typically 4x10-5 m/sec, or 0.04 mm/second! To go one meter at this speed takes about 10 hours! 9 Summary Current: Charge Displacement Drift speed I ΔQ = q(nA Δx) Δx = vdΔt Iavg=ΔQΔt= nqvdA4 10 Current Density J Let n = number of charged objects per unit volume q = charge of object = drift velocity of object The current density is current per unit area Generalization for many charged moving objects J ≡ nqvq⇒ niqivqii∑ vq J ≡ niqivqii∑11 Current and Current Density J J ≡ nqvq⇒ niqivqii∑Current is the flow (flux) of current density through an open surface Special case: uniform and perpendicular to surface J I =J ⋅ˆn dA =S∫J ⋅ dAS∫ I = JAP18- 12 Concept Question: Current Density A current I = 200 mA flows in the wire below. What is the magnitude of the current density J? 20 cm 10 cm 5 cm 1. J = 40 mA/cm 2. J = 20 mA/cm 3. J = 10 mA/cm 4. J = 1 mA/cm2 5. J = 2 mA/cm2 6. J = 4 mA/cm25 13 Conductivity and Resistivity σc: conductivity ρr: resistivity Ability of current to flow depends on density of charges & rate of scattering Two quantities summarize this: 14 Microscopic Ohm’s Law E =ρrJJ =σcE and depend only on the microscopic properties of the material, not on its shape ρr≡1σc ρrσc15 Demonstrations: Temperature Effects on Resistance F4 Conducting Glass F1 Conductivity of Ionizing Water F5 "http://tsgphysics.mit.edu/front/?page=demo.php&letnum=F%204&show=0 http://tsgphysics.mit.edu/front/?page=demo.php&letnum=F%205&show=0 http://tsgphysics.mit.edu/front/?page=demo.php&letnum=F%201&show=06 16 Why Does Current Flow? Instead of thinking of Electric Field, think of potential difference across the conductor 17 Ohm’s Law What is relationship between electric potential difference and current? ΔV = Vb− Va= −E ⋅ dsab∫= E J =Eρ=ΔV /ρJ =IA⎫⎬⎪⎪⎭⎪⎪⇒ ΔV = IρA⎛⎝⎜⎞⎠⎟≡ IR18 Drude Model Electrons scatter on average once every seconds. After every collision, direction of electron is random (hard sphere model) Between collisions, electric field E gives each electron a drift momentum If we average over all the electrons, then the initial velocities before the collision are random and add to zero so the average velocity after the collision is τ mevafter= mevbefore+ (−e)Eτ (vbefore)ave=0 vdrift≡ (vafter)ave= (−eτme)Ehttp://www.youtube.com/watch?v=dyX5I_io7bg7 19 Drude Model: Conductivity J = −nevdrift= (ne2τme)E=σcE ⇒σc=ne2τmehttp://www.youtube.com/watch?v=dyX5I_io7bg 20 Ohm’s Law R =ρA ΔV = IRR has units of Ohms (W) = Volts/Amp 21 How Big is an Ohm? • Short Copper Wire • Notebook paper (thru) • Typical resistors • You (when dry) • You (when wet) • Internally (hand to foot) milliohms (m ) ~1 G to 100 M 100 k 1 k 500 "Stick your wet fingers in an electrical socket: I = V / R 120 V / 1kΩ 0.1AYou’re dead! Ω Ω Ω Ω Ω Ω Ω8 P18- 22 Concept Question: Resistance When a current flows in a wire of length L and cross sectional area A, the resistance of the wire is 1. Proportional to A; inversely proportional to L. 2. Proportional to both A and L. 3. Proportional to L; inversely proportional to A. 4. Inversely proportional to both L and A Worked Ex.: Calculating Resistance 23 Consider a hollow cylinder of length L and inner radius a and outer radius b. The material has resistivity . Suppose a potential difference is applied between the ends of the cylinder and produces a current flowing parallel to the axis. What is the resistance measured? ρrWorked Ex.: Calculating Resistance 24 Consider a hollow cylinder of length L and inner radius a and outer radius b. The material has resistivity . When a potential difference is applied between the ends of the cylinder, current flows parallel to the axis. In this case, the cross-sectional area is and the resistance is given by ρr A =π(b2− a2) R =ρrLA=ρrLπ(b2− a2)9 Group Problem: Calculating Resistance 25 Consider a material of resistivity in a shape of a truncated cone of altitude h, and radii a and b, for the right and the left ends, respectively, as shown in the figure. Assuming that the current is distributed uniformly throughout the cross-section of the cone, what is the resistance between the two ends? You may find the following integral useful (where and are constants). ρr! du(αu +β)2∫= −1α(αu +β)26 Magnetic Fields 27 Magnetic Field of the Earth North magnetic pole located in southern hemisphere http://www.youtube.com/watch?v=AtDAOxaJ4Ms10 28
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