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How much current will flow in a circuit Analogy: Want to predict I Potential difference ∆V defines amount of the current that flows in a circuit • If water height difference (potential energy) is maintained, the flow of water will not change • The flow of water will depend on the water height difference Ohm’s law Ohm’s law Current through a conductor is proportional to ∆V: I∝ ∆V Notes: • This is not true for all conductors • Ohm’s law is not a universal law of physics (strictly speaking, it is not a law) Ohm’s law, however, is very useful since it is simple and works well for many commonly used materials. George Ohm 1789-1854 1 Resistance: definition The proportionality coefficient in Ohm’s law is called Resistance Definition of resistance ∆V R= I Units: Ohm, 1 Ω ≡ 1 V/A Ohm’s law: I = ∆V R Materials that obey Ohm’s law are called ohmic materials. Ohmic conductor: tungsten wire Non-ohmic materials: fluorescent light tube semicondutor diode Resistance: dependence on shape and size Resistance of a wire depends on its shape, size and properties of the material it is made of. Analogy: water pipes Which one has the highest resistance? - the one with the smallest area Which one has the highest resistance? - the one which is longer R∝ L A 2 Resistivity Resistance of a wire: R = ρ L A A L L - length A - cross-sectional area ρ - resistivity. Units: Ω.m • The resistivities of good conductors are small. Resistivities of some materials Silver Copper Aluminum Nichrome Carbon Conductors 1.59×10-8 Ω.m 1.67×10-8 Ω.m 2.65×10-8 Ω.m 108×10-8 Ω.m 3500×10-8 Ω.m Glass Rubber Wood Insulators 1010-1014 Ω.m 1013-1016 Ω.m 108-1011 Ω.m Resistivity: dependence on T Resistivity does not depend on size of shape, it is characteristic of the material Resistivity depends on two main factors: • the number of free charges per unit volume • the rate of collisions with ions The rate of collisions depends on the temperature T: higher T leads to higher ρ: ρ = ρ0 (1 + α∆T ) ρ0 - resistivity at temperature T0 ρ - resistivity at temperature T=T0+∆T α - temperature coefficient of resistivity This dependence is used in some thermometers See page 646 for values of α (p. 635 in old book) 3 “In series” and “in parallel” arrangement “In series” When two or more electrical devices are wired so that the same current flows through each one “In parallel” When two or more electrical devices are wired so that the potential difference across them is the same. Current splits between the devices at input and then joins back at output. Resistors in series 7 Resistors in series Loop: DABCD ∆V = 0 ∆VDA = VA - VD = ∆Vbattery = E =1.5 V ∆VAB = VB - VA = -IR1 ∆VBC = VC - VB = -IR2 ∆VCD = VD - VC = 0 1.5 V - IR1 -IR2 = 0 1.5 V - I(R1 + R2) = 0 Requivalent = R1 + R2 1.5 V - IReq = 0 For any number N of resistors connected in series Req = Ri = R1 +R2 + ... + RN Internal resistance of a battery Analogy: People can carry water only so fast, even if we remove the resistance the water flow will be limited by resistance in the battery 8 Internal resistance of a battery Real battery can be modeled as an ideal battery in series with a resistor r. r - internal resistance of a battery Part of voltage then drops on r. The voltage on battery terminals is: ∆V = E − Ir • ∆V on a load drops as current increases • Maximum current (Rload=0) is limited by r: Ishort circuit= E/r • With no load I=0, and ∆V on terminals is equal to emf: ∆V =E Batteries in series For any number N of batteries connected in series E eq = req = E i = E 1 +E 2 + ... + E N ri = r1 +r2 + ... + rN • Batteries are connected in series to increase emf. 9 Capacitors in series Q = C∆V Symbol: • Charge Q on all plates is the same ∆VAB = − Q C1 E = ∆VbC = − Q C2 ∆VcD = − Q C3 equivalent circuit 1 1 1 1 = + + Ceq C1 C2 C3 Q Q Q + + C1 C2 C3 E =Q 1 1 1 + + C1 C2 C3 E =Q 1 Q = Ceq Ceq For any number N of capacitors connected in series 1 = Ceq 1 1 1 1 = + + ... + Ci C1 C2 CN Resistors in parallel Analogy: equivalent resistance should be smaller than any of the resistances connected in parallel 10