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Properties of Current Current must flow in a complete circuit - current cannot be “lost” anywhere. Kirchoff’s point rule - the current flowing into any point or component must be equal to the current flowing out of it. Properties of Current Components in series must have the same current flowing through them. Properties of Current For components in parallel, the current is split between them. Water splitter analogy Properties of Current Components in series must have the same current flowing through them. For components in parallel, the current is split between them. Car tollgate analogy: Properties of Current Current must flow in a complete circuit - current cannot be “lost” anywhere. Kirchoff’s point rule - the current flowing into any point or component must be equal to the current flowing out of it. Components in series must have the same current flowing through them. For components in parallel, the current is split between them. Properties of p.d. The electric field is conservative so the potential at a given point is independent of the route taken. Kirchoff’s loop rule – the sum of potential differences around a closed loop must be zero. V4+V1=V3+V2 V3+V2+V1+V4=0 Gravitational analogy Properties of p.d. For a simple circuit, the sum of potential differences is equal and opposite to the e.m.f. of the power supply. The mechanism lifting marbles is analogous to e.m.f. The rest of the circuit is similar to the rolling down part of this toy. From the law of energy conservation: the work of the mechanism is equal to the energy the marble has on top. This is similar to the law that the e.m.f is equal to the potential difference in the circuit. Properties of p.d. If a circuit branches, the p.d. across each branch is the same and is not split. V2-V3=0 (from the red loop) emf=V1+V2 (from the blue loop) Similar to a waterfall: all water streams deliver the same energy Properties of p.d. The electric field is conservative so the potential at a given point is independent of the route taken. Kirchoff’s loop rule – the sum of potential differences around a closed loop must be zero. For a simple circuit, the sum of potential differences is equal and opposite to the e.m.f. of the power supply. If a circuit branches, the p.d. across each branch is the same and is not split. Resistors in Series and Parallel V1 I2 I1 V2 Why do resistances of two resistors in series add? A Rtotal R total V R1 R 2 Consider the first Kirchoff’s law for node A I2 I1 A For two resistors in series, the resistances add. I1=I2=I Consider the second Kirchoff’s law for the green loop V1+V2=V Resistors in Series and Parallel V1 I2 I1 A Rtotal R total V R1 R 2 V2 From Kirchoff’s Laws V1+V2=V I1=I2=I Ohm’s Law V1=I1R1=IR1; V2=I2R2=IR2 Since V1+V2=V we have: V=IR1+IR2=I(R1+R2) For two resistors in series, the resistances add. The same is true of three, four, five...resistances Ohm’s Law: Rtotal =V/I=I(R1+R2)/I Rtotal = R1+R2 Resistors in Series and Parallel I I1 I2 A B I1 I2 I Rtotal 1 1 1 R total R1 R 2 For two resistors in parallel, the resistances add in reciprocal to give the reciprocal resistance. Why do resistances of two resistors in parallel add in reciprocal? Consider the first Kirchoff’s law for node A: I=I1+I2 I I1 I2 A Resistors in Series and Parallel V1 I I1 I2 B A I1 I2 I V2 1 R total 1 1 Rtotal R1 R 2 For two resistors in parallel, the resistances add in reciprocal to give the reciprocal resistance. The same is true of three, four, five...resistances Why resistances of two resistors in parallel add in reciprocal? Consider the first Kirchoff’s law for node A: I=I1+I2 I I1 I2 A Consider the second Kirchoff’s law for the green loop V1-V2=0 V1=V2 Resistors in Series and Parallel V1 I I1 I2 From Kirchoff’s laws: B A I1 I2 I V2 1 R total 1 1 Rtotal R1 R 2 For two resistors in parallel, the resistances add in reciprocal to give the reciprocal resistance. The same is true of three, four, five...resistances I=I1+I2 ; V1=V2 =V From Ohm’s Law I1=V1 /R1= V/R1; I2=V2 /R2 =V/R2; Therefore, I=I1+I2= V/R1 + V/R2 = V(1/R1 +1/R2 ) From Ohm’s Law I= V/Rtotal 1 1 1 R total R 1 R 2 Resistors in Series and Parallel Rtotal 1 R total R 1 R 2 For two resistors in series, the resistances add. The same is true of three, four, five...resistances R total Rtotal 1 1 R1 R 2 For two resistors in parallel, the resistances add in reciprocal to give the reciprocal resistance. The same is true of three, four, five...resistances More complicated circuits 1/Rt1=1/56+1/33 = 0.018+0.03 = 0.048 1/Ohm Rt1=20.8 Ohm Rtotal=20.8 Ohm +47 Ohm = 67.8 Ohm Solving a complex resistor network E = 15V 2Ω A 9Ω A Point B 3Ω Point A 6Ω 15 Ω 3Ω 2Ω 10 Ω Point C A 5Ω Using Ohm’s Law and the rules for combining resistors, calculate the current flowing through the ammeters at points A, B and C Potential Divider Vout If a voltage supply is connected across two resistors in series, then we have a potential divider. V1 R1 in R2 V1 Vin V2 Potential Divider The current in the circuit I=Vin/Rtotal Since our resistors are in series, thus, Rtotal=R1+R2 From Ohm’s law: I I=Vin/(R1+R2) From Ohm’s law: Vout=V1=IR1; V2=IR2 The potential difference across each one is proportional to the resistance. V R V1 Vout in 1 R1 R2 Vout in Vin R2 V2 R1 R2 Even if we have a battery producing voltage Vin, we also can generate any voltage lower Vin by using a potential divider Battery Any real voltage source has an internal resistance, so whenever it is connected to a real load there is a potential divider effect. Battery Any real voltage source has an internal resistance, so whenever it is connected to a real load there is a potential divider effect. Internal resistance Ri and Resistance of a load are in series, thus, total resistance Rtotal=R+Ri Current in the circuit is I=E/(Rtotal) with E being the e.m.f of the battery Voltage produced by the battery is V=RI=E(R/Rtotal) Potential Divider If a voltage supply is connected across two resistors in series, then we have a potential divider. The potential difference across each one is proportional to the resistance. VR1 V1 R1 R 2 VR 2 V2 R1 R 2 Any real voltage source has an internal resistance, so whenever it is connected to a real load there is a potential divider effect.