Unit 4: Energy and power. Direct current circuits. Joule heating. Discharging process of a capacitor. DC circuits. Linear generator and receptor. Difference in potential between two points in a circuit. Pouillet’s law.
Joule heating Let’s consider a resistor (current I from a to b). Its terminals have potentials V a and V b (V a >V b ). Through a time dt the dQ=Idt charge moving from a to b is: I b a V b V a The energy lost by dQ on resistor going from a to b is: ( ) ( ) dU dQ V V Idt V V = − = − a b a b And the power (ratio of energy versus time): 2 dU V ( ) 2 P I V V IV I R = = − = = = a b dt R This energy is lost as heating in the conductor (Joule heating) due to the collisions between charges and atoms nuclei in the conductor. Tipler, chapter 25, section 25.3
Discharging process of a capacitor If a capacitor charged is connected to a resistor, the stored energy on capacitor is lost on resistor by Joule heating: dq ( t ) i ( t ) = − dt i(t) q(t) q ( t ) V 0 Q V ( t ) i ( t ) R = = C V(t) q ( t ) q ( t ) dq ( t ) dq ( t ) 1 From these equations: V ( t ) i ( t ) R R dt = = = − = − C C dt q ( t ) RC By integrating between t=0 (Q=CV 0 ) and a time t: q ( t ) t = dq ( t ) 1 q ( t ) t t − − RC RC dt q ( t ) V Ce V ( t ) V e − = = = 0 0 q ( t ) RC C CV 0 0 RC Time constant τ = V ( t ) 0 , 37 V V ( t 5 ) 0 , 007 V = τ = τ = = 0 0 Tipler, chapter 25, section 25.6
Direct current circuits. A simple circuit consists of a closed path (consisting of a conductor with no resistance) with some devices supplying power to the circuit (active devices) and others consuming power from the circuit (passive devices). Devices supplying power are called generators. Two kinds of devices consume power: Resistors: Turn electrical energy into heat by Joule heating. Receptors: Turn electrical energy into any kind of energy other than heat (mechanical, chemical, ……).
Direct current circuits. Introduction. Any circuit must always obey the energy conservation rule: Generated power=Consumed power A generator creates an electrical field in the circuit, thus enabling a steady D.C. + R ε - In some cases a generator can work as a receptor (a battery) M depending on the connection to the circuit.
Ideal generator. Emf The work done by the generator per unit of electrical charge passing dU through it is called electromotive ε = force (emf, ε ). unit: Volt. dq But the work done by unit of charge is V V − = ε a b the difference of potential (d.d.p.): The power generated by the dU dU dq P I = = = ε generator will be then: g dt dq dt Tipler, chapter 25, section 25.3
Real generator. Internal resistance In an Ideal Generator all the power generated (P g ) is supplied to the circuit (P s ), but what really happens is that some of this power is selfconsumed by the generator as Joule heating (P r ). It can be modelled by adding a resistor to the ideal generator to make it into a Real Generator. So: V V ε Ir − = − a b Real generator xI = (V V ) I εI I r 2 Internal resistance − = − a b + Ideal generator P P P = − s g r Tipler, chapter 25, section 25.3
Linear generator In a Generator the current must enter by the negative terminal and exit by the positive terminal. In this way the charges increase their electrical potential and can transfer energy to the receptors. V > V a b V V I r − = ε − a b V V − = ε r a b Ideal generator V V I r − = ε − a b Real generator ε and r: features of a linear generator
Linear receptor. Cemf A receptor turns electric energy into any kind of energy other than heat. (For example, an electric motor, an electrolitic cell, a charging battery…….). The energy turned in other than heat per unit of dU ' = electrical charge passing through the receptor is ε dq the contraelectromotive force (cemf, ε ’): dU dU dq The power turned out by the P t ' I = = = ε receptor will then be: dt dq dt + - I M Tipler, chapter 25, section 25.3
Linear receptor. Internal resistance As in generators, Joule heating also occurs in real receptors and is modelled through an internal resistor (r’): V V ε' Ir' − = + a b b xI ´ ε (V V ) I ε' I I r' 2 − = + - a b ' ε + a P P P = + c t r' P c is the power consumed by the receptor P t is the turned power by the receptor P r’ is the power lost as Joule heating by r’
Linear receptor In a receptor the current must enter by the positive terminal and exit by the negative terminal. In this way the charges lose their electrical potential and turn it into mechanical work, chemical energy, etc. V > V a b b ' V V I r' − = ε + ´ ε a b - ' ε + a V V ' r' I − = ε + b a V Real receptor ' ε V V ' − = ε r’ b a Ideal receptor ε ’ and r’ are the two features of a linear receptor I
Efficiency of generators and receptors The efficiency ( η ) of a generator is defined as the ratio between supplied power and generated power: P s η 1 = ≤ P g The efficiency ( η ’) of a receptor is defined as the rate of turned electrical power to consumed power: P t η' 1 = ≤ P c η and η ’ are dimensionless and they are measured as a %. η and η ’ are related to the energy lost as heating. For Ideal generators and receptors, η = η ’=1
Difference of potential between two points in a circuit The fall off of potential (d.d.p.) between two points in a circuit can be computed by adding the ddp of each device between A and B: ε, r ε ’ , r ’ R I 1 2 A B M - + - + ' V V I(R r r' ) − = + + − ε + ε A B
Difference in potential between two points in a circuit Another example: ε, r ε ’ , r ’ R 1 2 A I B M + - - + ' V V I(R r r' ) − = − + + + ε − ε A B
Difference in potential between two points in a circuit General rule ( ' ) V V RI − = − ε + ε A B We move along a path from A to B. I is positive if it goes from A to B. Negative if it goes from B to A. All resistors (R) between A and B are positive. Electromotive and contraelectromotive forces ( ε and ε ’) have the same sign as the terminal closest to B. BE CAREFUL: The direction to go from A to B is that determinig every sign. BE CAREFUL: The polarity of receptors must agree the direction of current. A receptor cannot work as a generator but a generator can work as a receptor (i.e a charging battery).
R Pouillet’s Law Let’s take a closed I circuit and a point A in ε ’ , r ’ M ε, r + this circuit and a - direction for I. The direction of the A choosen intensity is Pouillet’s law that determining every sign. ( ' ) ε + ε ( ' ) V V 0 I R I − = = − ε + ε = A A R BE CAREFUL: If I results negative: - If there isn’t any recepetor in the circuit, I equals the computed intensity but in the opposite direction. - If some receptor is in the circuit, we have to change the direction of I and recalculate it.
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