Current and Resistance

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Chapter 17

• Current and Resistance

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17.1 Electric Current

• Whenever electric charges move, an electric
  current is said to exist
• The current is the rate at which the charge flows
  through a certain cross-section
• For the current definition, we look at the
  charges flowing perpendicularly to a surface of
  area A

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Definition of the current

• Charge in motion through an area A. The time
  rate of the charge flow through A defines the
  current (charges per time)
• IDQ/Dt
• Units C/sAs/sA
• SI unit of the current Ampere

-
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Electric Current, cont

• The direction of current flow is the direction
  positive charge would flow
• This is known as conventional (technical) current
  flow, i.e., from plus () to minus (-)
• However, in a common conductor, such as copper,
  the current is due to the motion of the
  negatively charged electrons
• It is common to refer to a moving charge as a
  mobile charge carrier
• A charge carrier can be positive or negative

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17.2 Current and Drift Speed

• Charged particles move through a conductor of
  cross-sectional area A
• n is the number of charge carriers per unit
  volume V (concentration)
• nADxnV is the total number of charge carriers in
  V

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Current and Drift Speed, cont

• The total charge is the number of carriers times
  the charge per carrier, q (elementary charge)
• ?Q (nA?x)q unit (1/m3)(m2 m)AsC
• The drift speed, vd, is the speed at which the
  carriers move
• vd ?x/?t
• Rewritten ?Q (nAvd?t)q
• Finally, current, I ?Q/?t nqvdA

?x
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Current and Drift Speed, final

• If the conductor is isolated, the electrons
  undergo (thermal) random motion
• When an electric field is set up in the
  conductor, it creates an electric force on the
  electrons and hence a current

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Charge Carrier Motion in a Conductor

• The electric field force F imposes a drift on
  an electrons random motion (106 m/s) in a
  conducting material. Without field the electron
  moves from P1 to P2. With an applied field the
  electron ends up at P2 i.e., a distance vdDt
  from P2, where vd is the drift velocity
  (typically 10-4 m/s).

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Does the direction of the current depend on the
sign of the charge? No!
qvd
E
vd

• (a) Positive charges moving in the same
  direction of the field produce the same positive
  current as (b) negative charges moving in the
  direction opposite to the field.

E
vd
(-q)(-vd) qvd
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Current density

• The current per unit cross-section is called
  the current density J
• JI/A nqvdA/Anqvd
• In general, a conductor may contain several
  different kinds of charged particles,
  concentrations, and drift velocities. Therefore,
  we can define a vector current density
• Jn1q1vd1n2q2vd2
• Since, the product qvd is for positive and
  negative charges in the direction of E, the
  vector current density J always points in the
  direction of the field E.

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Example

• An 18-gauge copper wire (diameter 1.02 mm)
  carries a constant current of 1.67 A to a 200 W
  lamp. The density of free electrons is 8.5?1028
  per cubic meter. Find the magnitudes of (a) the
  current density and (b) the drift velocity.
•  

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Solution

• (a) Ad2p/4(0.00102 m)2p/48.2?10-7 m2
• JI/A1.67 A/(8.2?10-7 m2)2.0?106 A/m2
• (b) From JI/Anqvd, it follows

vd1.5?10-4 m/s0.15 mm/s
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17.3 Electrons in a Circuit

• The drift speed is much smaller than the average
  speed between collisions
• When a circuit is completed, the electric field
  travels with a speed close to the speed of light
• Although the drift speed is on the order of 10-4
  m/s the effect of the electric field is felt on
  the order of 108 m/s

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Meters in a Circuit Ammeter

• An ammeter is used to measure current
• In line with the bulb, all the charge passing
  through the bulb also must pass through the meter
  (in series!)

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Meters in a Circuit - Voltmeter

• A voltmeter is used to measure voltage (potential
  difference)
• Connects to the two ends of the bulb (parallel)

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Look at the four circuits shown below and
select those that will light the bulb.
QUICK QUIZ 17.2
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17.4 Resistance and Ohms law

• In a homogeneous conductor, the current density
  is uniform over any cross section, and the
  electric field is constant along the length.

b
a
VVa-VbEL
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Resistance

• The ratio of the potential drop to the current
  is called resistance of the segment
• Unit V/AW (ohm)

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Resistance, cont

• Units of resistance are ohms (O)
• 1 O 1 V / A
• Resistance in a circuit arises due to collisions
  between the electrons carrying the current with
  the fixed atoms inside the conductor

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Ohms Law

• V ? I ? Vconst.?I ? VRI
• Ohms Law is an empirical relationship that is
  valid only for certain materials
• Materials that obey Ohms Law are said to be
  ohmic
• IV/R
• R??, I?0, open circuit R?0, I??, short circuit

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Ohms Law, final
Ohmic

• Plots of V versus I for (a) ohmic and (b)
  nonohmic materials. The resistance RV/I is
  independent of I for ohmic materials, as is
  indicated by the constant slope of the line in
  (a).

Nonohmic