Unit 4: Energy and power. Direct current circuits. Joule heating. - - PowerPoint PPT Presentation

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

Tipler, chapter 25, section 25.3

• Let’s consider a resistor (current I from a to b). Its terminals

have potentials Va and Vb (Va>Vb). Through a time dt the charge moving from a to b is: The energy lost by dQ on resistor going from a to b is: And the power (ratio of energy versus time):

This energy is lost as heating in the conductor (Joule heating) due to the collisions between charges and atoms nuclei in the conductor.

dQ=Idt I a b Va Vb

) ( ) (

b a b a

V V Idt V V dQ dU − = − =

R V R I IV V V I dt dU P

b a 2 2

= = = − = = ) (

RC t RC t t ) t ( q CV

e V C ) t ( q ) t ( V Ce V ) t ( q dt RC 1 ) t ( q ) t ( dq

− −

= =  =  − = 



Discharging process of a capacitor

Tipler, chapter 25, section 25.6

• If a capacitor charged is connected to a resistor, the stored energy on

capacitor is lost on resistor by Joule heating: From these equations: By integrating between t=0 (Q=CV0) and a time t:

Q

dt ) t ( dq ) t ( i − =

i(t) q(t) V(t)

dt RC 1 ) t ( q ) t ( dq R dt ) t ( dq C ) t ( q R ) t ( i C ) t ( q ) t ( V − =        − =  = = R ) t ( i C ) t ( q ) t ( V = =

V0

RC = τ

Time constant

V 37 , ) t ( V = =τ V 007 , ) 5 t ( V = = τ

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
• ther than heat (mechanical, chemical, ……).

• +

ε R M

Direct current circuits. Introduction.

• Any circuit must always obey the energy conservation rule:

Generated power=Consumed power

• In some cases a generator can

work as a receptor (a battery) depending on the connection to the circuit.

• A

generator creates an electrical field in the circuit, thus enabling a steady D.C.

Ideal generator. Emf

• The work done by the generator

per unit of electrical charge passing through it is called electromotive force (emf, ε). unit: Volt.

dq dU = ε

Tipler, chapter 25, section 25.3

• The power generated by the

generator will be then:

dU dU dq P I dt dq dt

g

ε = = =

ε = −

b a

V V

• But the work done by unit of charge is

the difference of potential (d.d.p.):

Real generator. Internal resistance

• In an Ideal Generator all the power generated (Pg) is supplied

to the circuit (Ps), but what really happens is that some of this power is selfconsumed by the generator as Joule heating (Pr). It can be modelled by adding a resistor to the ideal generator to make it into a Real Generator. So:

Tipler, chapter 25, section 25.3

Real generator Ideal generator Internal resistance

= +

r

P P P

s g

= −

Ir ε V V

b a

− = −

xI

r I εI I ) V (V

b a 2

− = −

r I V V

b a

− = − ε

b a

V V >

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.

ε and r: features of a linear generator

r ε = −

b a

V V r I V V

b a

− = − ε

Ideal generator Real generator

• The energy turned in other than heat per unit of

electrical charge passing through the receptor is the contraelectromotive force (cemf, ε’):

Linear receptor. Cemf

dq dU '= ε

• A receptor turns electric energy into any kind of energy
• ther

than heat. (For example, an electric motor, an electrolitic cell, a charging battery…….).

I ' dt dq dq dU dt dU Pt ε = = =

• The power turned out by the

receptor will then be:

Tipler, chapter 25, section 25.3

M

• +

I

• As in generators, Joule heating also occurs in real receptors

and is modelled through an internal resistor (r’):

Linear receptor. Internal resistance

r' t c

P P P + =

Ir' ε' V V

b a

+ = −

xI

r' I I ε' I ) V (V

b a 2

+ = −

Pc is the power consumed by the receptor Pt is the turned power by the receptor Pr’ is the power lost as Joule heating by r’

ε

´

' ε

• +

a b

' ε − = +

a b

V V I r'

b a

V V >

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.

ε’ and r’ are the two features of a linear receptor

ε

´

' ε

• +

a b

Ideal receptor Real receptor

r’

' ε = −

a b

V V I r' V V

a b

+ = − ' ε

' ε

I V

• The efficiency (η’) of a receptor is defined as the rate of

turned electrical power to consumed power:

t c

P η' 1 P = ≤

Efficiency of generators and receptors

• 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
• The efficiency (η) of a generator is defined as the ratio

between supplied power and generated power:

P η 1 P

s g

= ≤

• 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:

R

• ε,r

+

M ε’,r’

I

' ε ε + − + + = − ) r' r I(R V V

B A

A B 1 2

+

• Difference of potential between two points in a circuit

• Another example:

R ε,r

• +

M ε’,r’

I

A B 1 2

' ε ε − + + + − = − ) r' r I(R V V

B A

• +

Difference in potential between two points in a circuit

• 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.

) ' (

 

+ − = − ε ε RI V V

B A Difference in potential between two points in a circuit General rule

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). BE CAREFUL: The direction to go from A to B is that determinig every sign.

• ε,r

+

ε’,r’

I

M R

Pouillet’s Law

• Let’s

take a closed circuit and a point A in this circuit and a direction for I.

• The direction of the

choosen intensity is that determining every sign.

A

   

+ =  + − = = − R I R I V V

A A

) ' ( ) ' ( ε ε ε ε

Pouillet’s law

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.