The two resistors in series can be replaced by one equivalent resistor R(Figure 7.4.1b) with the identical voltage drop =AV=IR that implies that Reg =R+R (7.4.2) The above argument can be extended to N resistors placed in series.The equivalent resistance is just the sum of the original resistances, R=R+R+=立R (7.4.3) Notice that if one resistance R is much larger than the other resistances R,then the equivalent resistance R is approximately equal to the largest resistor R. Next let's consider two resistors R and R,that are connected in parallel across a source of emf g,(Figure 7.4.2a). R W Reg Figure 7.4.2 (a)Two resistors in parallel.(b)Equivalent resistance By current conservation,the current that passes through the source of emf must divide into a current I that passes through resistor R and a current 1,that passes through resistor R2.Each resistor individually satisfies Ohm's law,AV=IR and AV2=12 R. However,the potential across the resistors are the same,AV=AV,=g.Current conservation then implies (7.4.4) The two resistors in parallel can be replaced by one equivalent resistor R with e=IR (Figure 7.4.2b).Comparing these results,the equivalent resistance for two resistors that are connected in parallel is given by 7-11
7-11 The two resistors in series can be replaced by one equivalent resistor Req (Figure 7.4.1b) with the identical voltage drop ε = ΔV = I Req that implies that Req = R1 + R2 . (7.4.2) The above argument can be extended to N resistors placed in series. The equivalent resistance is just the sum of the original resistances, eq 1 2 1 N i i R R R R = = + + = ∑ . (7.4.3) Notice that if one resistance R1 is much larger than the other resistances Ri , then the equivalent resistance Req is approximately equal to the largest resistor R1. Next let’s consider two resistors R1 and R2 that are connected in parallel across a source of emf ε , (Figure 7.4.2a). Figure 7.4.2 (a) Two resistors in parallel. (b) Equivalent resistance By current conservation, the current I that passes through the source of emf must divide into a current I 1 that passes through resistor R1 and a current I2 that passes through resistor R2. Each resistor individually satisfies Ohm’s law, ΔV1 = I 1 R1 and ΔV2 = I2 R2 . However, the potential across the resistors are the same, ΔV1 = ΔV2 = ε . Current conservation then implies I = I 1 + I2 = ε R1 + ε R2 = ε 1 R1 + 1 R2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ . (7.4.4) The two resistors in parallel can be replaced by one equivalent resistor Req with ε = IReq (Figure 7.4.2b). Comparing these results, the equivalent resistance for two resistors that are connected in parallel is given by
111 (7.4.5) Reg R R2 This result easily generalizes to N resistors connected in parallel 1_1+1+1+=】 (7.4.6) Reg R R2 R3 When one resistance R is much smaller than the other resistances R,then the equivalent resistance R is approximately equal to the smallest resistor R.In the case of two resistors, RR:-RR--R Reg=R+R R2 This means that almost all of the current that enters the node point will pass through the branch containing the smallest resistance.So,when a short develops across a circuit,all of the current passes through this path of nearly zero resistance 7.5 Kirchhoffs Circuit Rules In analyzing circuits,there are two fundamental(Kirchhoff's)rules. Junction Rule At any point where there is a node formed by the junction of various current carrying branches,by current conservation,the sum of the currents into the node must equal the sum of the currents out of the node (otherwise charge would build up at the junction); ∑1n=∑1out (7.5.1) As an example,consider Figure 7.5.1 below: Figure 7.5.1 Kirchhoff's junction rule. According to the junction rule,the three currents are related by 7-12
7-12 eq 1 2 1 1 1 R R R = + . (7.4.5) This result easily generalizes to N resistors connected in parallel eq 1 2 3 1 1 1 1 1 1 N i R R R R = Ri = + + + = ∑ . (7.4.6) When one resistance R1 is much smaller than the other resistances Ri , then the equivalent resistance Req is approximately equal to the smallest resistor R1 . In the case of two resistors, 1 2 1 2 eq 1 1 2 2 R R R R R R R R R = ≈ = + . This means that almost all of the current that enters the node point will pass through the branch containing the smallest resistance. So, when a short develops across a circuit, all of the current passes through this path of nearly zero resistance. 7.5 Kirchhoff’s Circuit Rules In analyzing circuits, there are two fundamental (Kirchhoff’s) rules. Junction Rule At any point where there is a node formed by the junction of various current carrying branches, by current conservation, the sum of the currents into the node must equal the sum of the currents out of the node (otherwise charge would build up at the junction); in out ∑I = ∑I . (7.5.1) As an example, consider Figure 7.5.1 below: Figure 7.5.1 Kirchhoff’s junction rule. According to the junction rule, the three currents are related by
1=12+13 Loop Rule The sum of the voltage drops Al,across any circuit elements that form a closed circuit is zero: △V=0. (7.5.2) closed loop The rules for determining Al across a resistor and a battery with a designated travel direction are shown below: travel direction travel direction higher V lower V lowerV higher V W W a △V=b-Va=-IR b a b △V='%-'a=+R travel direction travel direction e lower V higher V higher V lower V b Q b △V='b-'a=+e △V='b-'a=-e Figure 7.5.2 Rules for determining potential difference across resistors and batteries. Note that the choice of travel direction is arbitrary.The same equation is obtained whether the closed loop is traversed clockwise or counterclockwise. Example 7.5.1:Voltage divider Consider a source of emf =V that is connected in series to two resistors,R and R R out Figure 7.5.3 Voltage divider. The potential difference,Vo,across resistor R2 will be less than V.This circuit is called a voltage divider.From the loop rule, 7-13
7-13 1 2 3 I = I + I . Loop Rule The sum of the voltage drops ΔV , across any circuit elements that form a closed circuit is zero: closed loop ∑ ΔV = 0 . (7.5.2) The rules for determining ΔV across a resistor and a battery with a designated travel direction are shown below: Figure 7.5.2 Rules for determining potential difference across resistors and batteries. Note that the choice of travel direction is arbitrary. The same equation is obtained whether the closed loop is traversed clockwise or counterclockwise. Example 7.5.1: Voltage divider Consider a source of emf ε = Vin that is connected in series to two resistors, R1 and R2 Figure 7.5.3 Voltage divider. The potential difference, Vout , across resistor R2 will be less than Vin . This circuit is called a voltage divider. From the loop rule
Vin-IR-IR2 =0. (7.5.3) Therefore the current in the circuit is given by I=- Vin (7.5.4) R+R2 Thus the potential difference,Vo,across resistor R2 is given by =R=R。 (7.5.5) R+R Note that the ratio of the potential differences characterizes the voltage divider and is determined by the resistors: R (7.5.6) VR+R 7.6 Voltage-Current Measurements Any instrument that measures potential difference or current will disturb the circuit under observation.In some devices,known as ammeters,the current in a coil will cause meter movement (arising from the torque on a magnetic dipole in an magnetic field,a topic will soon study)or some change will result in a digital display.There will be some potential difference due to the resistance of the current through the ammeter.An ideal ammeter has zero resistance.However in the case of an ammeter that has resistance of 1 on the 250 mA range.The drop of 0.25 V may or may not be negligible;knowing the meter resistance allows one to correct for its effect on the circuit. An ammeter can be converted to a voltmeter by putting a resistor R in series with the coil movement.The potential difference across some circuit element can be determined by connecting the coil movement and resistor in parallel with the circuit element.This causes a small amount of current to flow through the coil movement.The voltage drop across the element can now be determined by measuring and computing the voltage drop from AV =IR,which is read on a calibrated scale.The larger the resistance R,the smaller the amount of current is diverted through the coil.Thus an ideal voltmeter would have an infinite resistance. Resistor Value Chart 0 Black 4 Yellow 8 Gray 1 Brown 5 Green 9 White 2 Red 6 Blue -1 Gold 3 Orange 7 Violet -2 Silver 7-14
7-14 in 1 2 V − IR − IR = 0 . (7.5.3) Therefore the current in the circuit is given by in 1 2 V I R R = + (7.5.4) Thus the potential difference, Vout , across resistor R2 is given by 2 out 2 in 1 2 R V IR V R R = = + . (7.5.5) Note that the ratio of the potential differences characterizes the voltage divider and is determined by the resistors: Vout Vin = R2 R1 + R2 (7.5.6) 7.6 Voltage-Current Measurements Any instrument that measures potential difference or current will disturb the circuit under observation. In some devices, known as ammeters, the current in a coil will cause meter movement (arising from the torque on a magnetic dipole in an magnetic field, a topic will soon study) or some change will result in a digital display. There will be some potential difference due to the resistance of the current through the ammeter. An ideal ammeter has zero resistance. However in the case of an ammeter that has resistance of 1Ω on the 250 mA range. The drop of 0.25 V may or may not be negligible; knowing the meter resistance allows one to correct for its effect on the circuit. An ammeter can be converted to a voltmeter by putting a resistor R in series with the coil movement. The potential difference across some circuit element can be determined by connecting the coil movement and resistor in parallel with the circuit element. This causes a small amount of current to flow through the coil movement. The voltage drop across the element can now be determined by measuring I and computing the voltage drop from ΔV = IR , which is read on a calibrated scale. The larger the resistance R, the smaller the amount of current is diverted through the coil. Thus an ideal voltmeter would have an infinite resistance. Resistor Value Chart 0 Black 4 Yellow 8 Gray 1 Brown 5 Green 9 White 2 Red 6 Blue −1 Gold 3 Orange 7 Violet −2 Silver