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Chapter 20

Circuits

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1) Electric current and emf

a) Potential difference and charge flowBattery produces potential difference causing flow of charge in conductor

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b) Current: I = q/t

∆ q is charge that passes the surface in time ∆ t

Units: C/s = ampere = A

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• Drift velocity: average velocity of electrons

~ mm/s

• Signal velocity: speed of electric field

= speed of light in the material ~108 m/s

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• emf = electromotive force = maximum potential difference produced by a device

• Symbol: E• emf is not a force, but it causes current to flow

Eis like gh

c) Electromotive force, emf

gravitational analogy for a circuit

battery

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• Symbol for a perfect seat of emf

E

V = E

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• Real battery

r

R

V < Ein general

Battery terminals

E

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2) Ohm’s Law

• Ohm’s law: for some devices (conductors), I is proportional to V:

IV Device

I

V

V = IR

• R = Resistance = proportionality constant = V/I

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• Current depends on voltage

IV Device

I

V

I

V V

I

and on the device

• Resistance R = V / I, not necessarily constant

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• Ohmic material obeys Ohm’s Law: R is constant• R is a property of the device

IV Device

• symbol:

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3) Resistivity

• Property of material; zero for superconductors

• For cylindrical conductor:

• R is proportional to L• R is proportional to 1/A• R is proportional to L / A• Define resistivity as the proportionality constant

€

R = ρL

A

a) Definition

A

L

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b) values

• Conductors: ~ 10-8 m (Cu, Ag best)

• Semiconductors: ~ 1 - 103 m (Ge, Si)

• Insulators: ~ 1011 - 1016 m (rubber, mica)

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c) Temperature dependence

• Resistivity is linear with temperature:

For metals, > 0 (resistance increases with temp)For semiconductors, < 0 (resistance decreases)

€

=a + bT

€

=0 + b(T −T 0)

€

0 = resistivity at T = T0

€

/ρ0 = 1 + α (T −T 0)€

a + bT0

€

= coefficient of resistivity (C º -1 )

€

=0(1+ α (T −T 0))

€

⇒ R = R0(1 + α (T −T 0))

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d) Superconductors• Below critical temp Tc, –> 0

– Current flows in loop indefinitely– Quantum transitions not possible

Tc typically < 10 K, but can be > ~ 75 K (high Tc ceramics) (record is 138 K)

Applications: MRI, MagLev trains

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4) Power and Energy

• Energy lost or gained by q is UqV

• Power:

V I

€

P =ΔU

Δt

Units: (C/s)(J/C) = J/s = WConsumed energy = P t: [kW h] = (1000 W) (3600 s) = 3.6 MJ

€

=qV

Δt

€

P = VI

a) Power dissipated in a device

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b) Power dissipated in resistors

V I V = IR

€

P = VI

€

=(IR)I

€

⇒

€

P = I2R

€

P = VI

€

=VV

R

€

⇒

€

P =V 2

R

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6) AC/DC

a) Direct (Constant) Current

IV

V

t

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b) Alternating Current

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It

V0

-V0

V

ac generator alternates polarity:

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e.g. V = V0 sin(ωt)

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Average voltage: zero

€

Vrms = V 2 =V0

2t

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

V0

-V0

V

t

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I0

-I0

I

Average current: zero

€

Irms = I2 =I0

2

Average power:

€

P = 12 V0I0

For resistors

€

=VrmsIrms

€

P =V0

2

I0

2

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6) Circuit wiring

a) Basic circuit

IE

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b) Ground

One point may be referred to as ground

IE

The ground may be connected to “true” ground through water pipes, for example.

IE=

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d) Open circuit

EI

c) Short circuit

E

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f) Parallel connection

e) Series connection

same currentI

same voltageV

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7) Resistors in series

For perfect conductors

€

V = V1 + V2

From Ohm’s law

€

V1 = IR1 and V2 = IR2

€

So, V = IR1 + IR2

€

=I(R1 + R2)

€

Or, V = IRS

€

RS = R1 + R2if

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In general, for series resistors,

€

RS = R1 + R2 + R3 +L

€

RS = Rii

∑

Find the current and the power through each resistor.

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Voltage divider

€

I =V

RS

=V

R1 + R2

V=10VR1=6

R2=4

€

=1A

Current is the same in both resistors

I

Voltages divide in proportion to R

€

V1 = IR1 = 6V

€

V2 = IR2 = 4VVo

Output Voltage:

€

Vo = IR2

€

=VR2

R1 + R2

⎛

⎝ ⎜ ⎞

⎠ ⎟

€

=V

R1 + R2

R2

V

Vo

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8) Resistors in parallel

a) General caseConservation of charge

€

I = I1 + I2

Ohm’s Law

€

V = I1R1 and V = I2R2

€

So, I =V

R1

+V

R2

€

=V1

R1

+1

R2

⎛

⎝ ⎜ ⎞

⎠ ⎟

€

Or, V = IRP

€

1

RP

=1

R1

+1

R2

if

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€

1

RP

=1

R1

+1

R2

€

RP =R1R2

R1 + R2

€

=R1 //R2

• Equivalent resistance is smaller than either R1 or R2

• Conductance adds

In general, for parallel resistors,

€

1

RP

=1

R1

+1

R2

+1

R3

+L

or

€

1

RP

=1

R ii∑

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conductance adds

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parallel connections in the home

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b) Special cases

i) Equal resistance

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RP = R //R =R2

2R

€

=R

2

ii) Very unequal resistors (e.g. 1 and 1 M

€

RP = R1 //R2 =R1R2

R1 + R2

€

=(1)(106)Ω

1 +106 ≅ 1Ω

€

If R2 >> R1, then R1 + R2 ≅ R2

€

so RP ≅R1R2

R2

= R1 RP = the smaller value