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. V 4 +V 1 =V 3 +V 2 V 3 +V 2 +V 1 +V 4 =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 For two resistors in series, the resistances add. R total V I1I1 I2I2 A  Why do resistances of two resistors in series add?  Consider the first Kirchoff’s law for node A  Consider the second Kirchoff’s law for the green loop I1I1 I2I2 A I 1 =I 2 =I V1V1 V2V2 V 1 +V 2 =V

Resistors in Series and Parallel For two resistors in series, the resistances add. The same is true of three, four, five...resistances R total V I1I1 I2I2 A  From Kirchoff’s Laws  Ohm’s Law V 1 =I 1 R 1 =IR 1 ; V 2 =I 2 R 2 =IR 2  Since V 1 +V 2 =V we have: V=IR 1 +IR 2 =I(R 1 +R 2 )  Ohm’s Law: R total =V/I=I(R 1 +R 2 )/I  R total = R 1 +R 2 V1V1 V2V2 V 1 +V 2 =V I 1 =I 2 =I

Resistors in Series and Parallel For two resistors in parallel, the resistances add in reciprocal to give the reciprocal resistance. R total  Why do resistances of two resistors in parallel add in reciprocal?  Consider the first Kirchoff’s law for node A: I=I 1 +I 2 I2I2 A I1I1 I I1I1 I2I2 B I A I1I1 I2I2 I

Resistors in Series and Parallel For two resistors in parallel, the resistances add in reciprocal to give the reciprocal resistance. The same is true of three, four, five...resistances R total  Why resistances of two resistors in parallel add in reciprocal?  Consider the first Kirchoff’s law for node A: I=I 1 +I 2  Consider the second Kirchoff’s law for the green loop I2I2 A I1I1 I I1I1 I2I2 B I A I1I1 I2I2 I V1V1 V2V2 V 1 -V 2 =0 V 1 =V 2

Resistors in Series and Parallel For two resistors in parallel, the resistances add in reciprocal to give the reciprocal resistance. The same is true of three, four, five...resistances R total  From Kirchoff’s laws: I=I 1 +I 2 ; V 1 =V 2 =V  From Ohm’s Law I 1 =V 1 /R 1 = V/R 1 ; I 2 =V 2 /R 2 =V/R 2 ;  Therefore, I=I 1 +I 2 = V/R 1 + V/R 2 = V(1/R 1 +1/R 2 )  From Ohm’s Law I= V/R total I2I2 A I1I1 I I1I1 I2I2 B I V1V1 V2V2

Resistors in Series and Parallel For two resistors in series, the resistances add. The same is true of three, four, five...resistances For two resistors in parallel, the resistances add in reciprocal to give the reciprocal resistance. The same is true of three, four, five...resistances R total

More complicated circuits 1/R t1 =1/56+1/33 = = /Ohm R t1 =20.8 Ohm R total =20.8 Ohm +47 Ohm = 67.8 Ohm

E = 15V 9 Ω 15 Ω 2 Ω 3 Ω 2 Ω 10 Ω 5 Ω 6 Ω 3 Ω Using Ohm’s Law and the rules for combining resistors, calculate the current flowing through the ammeters at points A, B and C Point A Point B Point C A A A Solving a complex resistor network

If a voltage supply is connected across two resistors in series, then we have a potential divider. Potential Divider V1V1 V in in V out V2V2 V1V1 R2R2 R1R1

The current in the circuit I=V in /R total Since our resistors are in series, thus, R total =R 1 +R 2 From Ohm’s law: I=V in /(R 1 +R 2 ) From Ohm’s law: V out =V 1 =IR 1 ; V 2 =IR 2 The potential difference across each one is proportional to the resistance. Potential Divider Even if we have a battery producing voltage V in, we also can generate any voltage lower V in by using a potential divider in V out I

Any real voltage source has an internal resistance, so whenever it is connected to a real load there is a potential divider effect. Battery

Internal resistance R i and Resistance of a load are in series, thus, total resistance R total =R+R i Current in the circuit is I= E /(R total ) with E being the e.m.f of the battery Voltage produced by the battery is V=RI= E (R/R total ) Battery

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. Any real voltage source has an internal resistance, so whenever it is connected to a real load there is a potential divider effect. Potential Divider