Chapter 24: Electric Current

•  Electric current is a net flow of electric charge. •  Quantitatively, current is the rate at which charge crosses a

given area.

•  n is the number of charge carriers per unit volume, each with charge q. •  vd is the drift velocity, or average velocity of the charge carriers along

the direction of the current. •  q is the charge of each charge carrier (which may be positive or

negative)

Electric current

I =dQ

dt

I = q n A vd

dQ = q(n A L) = q(n A vd dt)

Clicker question •  Which one of the following represents a non-zero

current flowing from right to left? A.  A beam of electrons moves from left to right. B.  In a solution, positive ions move right and negative ions move

left. C.  Blood, carrying positive and negative ions at the same speed,

moves from right to left through a vein. D.  A metal car with no net charge speeds to the left. E.  More than one of the above has a current flowing from right to

left.

Current density •  Current density, J , is the current per unit area.

•  Current density may vary with position in both direction and magnitude.

•  In general, the current through an area is the flux of the current density over that area:

•  When current density is uniform and perpendicular to a flat area, current becomes the product of current density with area: I = JA

�J = n q �vd

I =

��J · �dA

Clicker question CT 29.2a

A copper cylinder is machined to have the following shape. The ends are connected to a battery so that a steady, constant current flows through the copper.

Which region has the greatest current density |J|?

A B C

A, B, C, or D: All three are the same E: Not sure/not enough info?

Conduction in metals (a classical model) •  A metal contains a “sea” of free electrons (weakly bound valence electrons): •  They’re confined to the metal but not attached to individual atoms. •  The electrons move about in random directions with high thermal velocities.

•  If no electric field, there’s no current associated with thermal motion.

•  Applying an electric field superposes a small drift velocity on the electrons’ motion. •  Between collisions electron

accelerates due to electric field: •  Average velocity due to this

acceleration is drift velocity: •  Microscopic Ohm’s “law”:

•  τ is the average time between collisions.

•  σ is conductivity •  Often σ is independent of electric

field (property of metal and T)

�a =(−e) �E

me

�J = nq�vd =ne2τ

me

�E = σ �E

�vavg = �vd ≈ aτ = − eτ

me

�E

E is nonzero inside conductor!?

Ohm’s law •  The microscopic version of Ohm’s law relates the current density at

a point in a conductor to the electric field at that point: (ρ is the resistivity) •  Consider a particular piece of conducting material with uniform

cross section. How does current relate to voltage across the conductor?

•  Note: V represents magnitude of

potential difference across resistor. (voltage always decreases along current)

•  R is resistance (depends on both material and geometry) •  We will assume wires in circuits, attached to resistors and other components,

have negligible resistance

�J = σ �E =�E

ρ

E = ρJV (= |∆V |) = EL = ρLJ = I

�ρL

A

�= IR

Clicker question •  The figure shows three pieces of wire. Wires (1) and (2)

are made from the same material, while (3) is made from a material with twice the resistivity. Wires (1) and (3) have twice the diameter of (2). If the same voltage is applied across each, which will pass the largest current?

A.  Wire (1) B.  Wire (2) C.  Wire (3)

CT 29.3

Two cylindrical resistors are made of the same material (same resistivity).

A) 2 B) 4 C) 1/2 D) 1/4 E) 1

1 2Resistor 2 is twice as long and has twice

the diameter of resistor 1. What is the ratio R2/R1?

Example •  Extension cords are often made from 18-gauge copper

wire (diameter 1.0 mm). An electric saw that draws 7.0 A is operated at the end of an 8.0 m long extension cord. Find the potential difference between the wall outlet and the saw. (ρcopper=1.7 x 10-8 Ω m).

Electromotive “Force” – EMF (Ch. 25.1) •  Where does the electric field in a current-carrying conductor come

from? •  an emf pumps charge from a lower to higher potential energy •  Batteries, electric generators, fuel cells, solar cells, are examples of

sources of emf.

CT 29.2b

A copper cylinder is machined to have the following shape. The ends are connected to a battery so that a current flows through the copper.

Which region has the greatest magnitude of current, I?

A B C

A, B, C, or D: All three are the same E: Not sure/not enough info?

CT 29.9 Two identical resistors are wired in series (one behind the other). They are attached to a battery a current is flowing through the circuit.

The current in the second resistor is _____ the current in the first resistor

A: Equal to B: Half C: Smaller than, but not necessarily half

CT 29.2d

A copper cylinder is machined to have the following shape. The ends are connected to a battery so that a current flows through the copper.

Which region has the greatest magnitude electric field |E|?

A B C

A, B, C, or D: All three are the same E: Not sure/not enough info?

Electric power •  A charge dq loses potential energy as it goes from one terminal of the

battery to the other:

•  Where does the energy go? •  dissipated as thermal energy in resistor •  emf source does positive work to bring

charge back to positive terminal. •  If circuit has a current I, what is rate

of work done by emf source?

•  P must be equal to rate of energy dissipation in resistor: Pdiss = VI = I2R

∆U = dq (−Vemf )

P =dW

dt=

dq

dtVemf = VemfI

Question 32.14 Lightbulbs

Two lightbulbs operate at 120 V, but one has a power rating of 25 W while

the other has a power rating of 100 W.

Which one has the greater resistance?

1) the 25 W bulb

2) the 100 W bulb

3) both have the same

4) this has nothing to do with resistance

CT 29.6

US Bulbs are rated for 120 V. European bulbs are designed for 240 V. If you buy a 100 W light bulb as a souvenir in Paris, and plug it in at home, what happens?

A: Glows as usual, like a 100 W bulb should B: Bzzzt, it burns out, too much power C: Glows half as bright (powerful) as usual D: It glows 1/4 as bright as usual E: None of these/???

CT 29.6b

Household appliances all have V=120 V in the US. What is the approximate resistance of a filament (at operating temperature) of a 100 W lightbulb?

A) R = 100 Ohms B) R = 144 Ohms C) R = 1250 Ohms D) None of the Above