Ch 192 Resistors in Series We want to find the single resistance R eq that has the same effect as the three resistors R 1, R 2, and R 3. Note that the current I is the same throughout the circuit since charge can’t accumulate anywhere. V is the voltage across the battery and also V = V 1 + V 2 + V 3 Since V 1 = I R 1 etc., we can say The equivalent equation is V=IR eq and thus
Ch 193 Resistors in Parallel This is called a parallel circuit Notice V 1 = V 2 = V 3 Since charge can’t disappear, we can say I = I 1 + I 2 + I 3 We can combine these equations with V = IR eq to give
Ch 196 EMF Devices that supply energy to an electric circuit are referred to as a source of electromotive force. Since this name meaningless, we just refer to them as source of emf (symbolized by and a slightly different symbol in the book.) Sources of emf such as batteries often have resistance which is referred to as internal resistance.
Ch 197 Terminal Voltage r ab Terminal Voltage V ab We can treat a battery as a source of in series with an internal resistor r. When there is no current then the terminal voltage is V ab = But with current I we have: The internal resistance is small but increases with age.
Ch 198 Kirchhoff’s Junction Rule Kirchhoff’s Rules are necessary for complicated circuits. Junction rule is based on conservation of charge. Junction Rule: at any junction, the sum of all currents entering the junction must equal the sum of all currents leaving the junction. I3I3 I2I2 22 I1I1 11 a b R1R1 R2R2 R3R3
Ch 199 Kirchhoff’s Junction Rule Kirchhoff’s Rules are necessary for complicated circuits. Junction rule is based on conservation of charge. Junction Rule: at any junction, the sum of all currents entering the junction must equal the sum of all currents leaving the junction. I3I3 I2I2 22 I1I1 11 a b R1R1 R2R2 R3R3 Point a: I 1 + I 2 = I 3 Point b: I 3 = I 1 + I 2
Ch 1910 Kirchhoff’s Loop Rule Loop rule is based on conservation of energy. Loop Rule: the sum of the changes in potential around any closed path of a circuit must be zero. I3I3 I2I2 22 I1I1 11 a b R1R1 R2R2 R3R3
Ch 1911 Kirchhoff’s Loop Rule Loop rule is based on conservation of energy. Loop Rule: the sum of the changes in potential around any closed path of a circuit must be zero. I3I3 I2I2 22 I1I1 11 a b R1R1 R2R2 R3R3 All loops clockwise: Upper Loop: + 2 – I 2 R 3 – I 3 R 1 – I 3 R 2 = 0 Lower Loop: + 1 + I 2 R 3 – 2 = 0 Large Loop: + 1 – I 3 R 1 – I 3 R 2 = 0
Ch 1912 Using Kirchhoff’s Rules Current: Current is the same in between junctions. Assign direction to current arbitrarily. If result is a negative current, it means that the current actually flows in the opposite direction. Don’t change direction, just give negative answer. Branches with a capacitor have zero current. Signs for Loop Rule Go around loop clockwise or counterclockwise. IR drop across resistor is negative if you are moving in direction of the current. Voltage drop across battery or other emf is positive if you move from minus to plus. Simultaneous Equations You will need one equation for each unknown. It pays to generate “extra” equations because they may lead to a simpler solution.
Ch 1914 Capacitors in Parallel V is the same for each capacitor The total charge that leaves the battery is Q = Q 1 + Q 2 + Q 3 = C 1 V + C 2 V + C 3 V Combine this with Q = C eq V to give:
Ch 1915 Capacitors in Series The charge on each capacitor must be the same. Thus Q = C 1 V 1 = C 2 V 2 = C 3 V 3 Combine this with V = V 1 + V 2 + V 3 to give:
Ch 1916 Charging a Capacitor When switch is closed, current flows because capacitor is charging As capacitor becomes charged, the current slows because the voltage across the resistor is - V c and V c gradually approaches . Once capacitor is charged the current is zero.
Ch 1917 RC Decay If a capacitor is charged and the switch is closed, then current flows and the voltage on the capacitor gradually decreases. Since I V C we can say that: It is necessary to use calculus to find:
Ch 1918 Exponential Decay The value = RC is called the time constant of the decay. If R is in and C is in F, then has units of seconds. During each time constant, the voltage falls to 0.37 of its original value. We can also define the half-life ( 1/2 ) by 1/2 = 0.693 RC. During each half-life, the voltage falls to ½ of its original value.
Ch 1920 Electric Hazards A current greater than 70 mA through the upper torso can be lethal. Wet skin: I = 120 V / 1000 = 120 mA Dry skin: I = 120 V / 10000 = 12 mA Your body can act as capacitor in parallel with resistor and this gives greater current for ac. The key to safety is don’t let your body become part of the circuit. Standing in water can give path to ground which will complete circuit. Metal cabinet grounded by 3-prong plug protects if there is loose wire inside because it causes short that trips circuit breaker. “Ground fault detector” should turn off current in time to protect you
Ch 1921 Ammeters and Voltmeters Ammeter To measure current it must be in circuit. Must have small internal resistance or it will reduce current and confuses measurement. Voltmeter To measure voltage difference, it must be connected to two different parts of circuit. Must have high internal resistance or it will draw too much current which reduces voltage difference and confuses measurement.