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Ch – 28 Current and Conductivity

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- 18th century: Franklin and others developed the single fluid theory of electricity. Electricity flows from a body with an excess of charge (positive) to one with a deficit of charge (negative).
- Theories of electricity advanced with the assumption of a positive charge carrier.
- 19th century: Thompson and others suggested that negatively-charged electrons were the charge carriers, in a conductor.
- Confirmed by Tolson and Stewart in 1916
- Most engineering applications still assume positive charge carriers (aka electron holes)

- Sea of electrons move randomly but net motion=0 when conductor is in equilibrium
- a force due to the presence of an external E field will cause the sea of electrons to move with vd - drift velocity

Moving sea of electrons

- The number of electrons per second that pass through a cross sectional area of wire or other conductor:
i = Ne/∆t

Ne (number of electrons) = i ∆t

Ne = nV where V is the volume of the wire (A ∆x) and n is the conduction electron density (on the order of 1028 electrons per m3)

∆x = vd ∆t, therefore:

Ne = nAvd ∆t

Ne/∆t = i = nAvd

- These four wires are made of the same metal. Rank, in order, from largest to smallest, the electron currents ia to id

i is proportional to r2vd. Changing r has more influence than changing v by the same amount

1.0 x 1016 electrons flow through a x-section of silver wire in 320 μs with a drift velocity of 8.0 x 10 -4 m/s. What is the diameter of the wire?

9.26 x 10-4 m

- The drift velocity of electrons is the same throughout the wire
- The electrons themselves can’t go anywhere while traveling through the wire
- Therefore the current going in is equal to the current coming out

An electron current is a non-equilibrium motion of charges sustained by an electric field

- On-axis field for charged ring
- points away from positive charge, towards ring for negative charge
- is proportional to the charge on the ring
- decreases with distance from the ring

d>b>e>a=c

E depends on the difference in the charge on the two rings. The E fields of a and c are zero. The difference is the greatest for the rings of d.

- What is the surface charge density of a 1.0 mm-diameter wire with 1000 excess electrons per cm of length?

ή = 5.1 x 10-12 C/m2

- The rate in coulombs per second, at which charge moves in the direction of E
- For constant current I = ∆Q/∆t
- For changing current I = dQ/dt

- Current direction from positive terminal to negative terminal, opposite the direction of electrons in a metal
- I = ∆Q/∆t = -(eNe/ ∆t) = -ei (sign for direction)

In an ionic solution, 5.0 x 1015 positive ions with charge +2e pass to the right, while 6.0 x 1015 negative ions with charge –e pass to the left. What is the current in the solution and what is the direction of the E field?

2.56 mA (milliamps).

E field is to the right

The current in a wire at time t is given by the expression: I = (2.0 A)e-t/(2.0μs) where t is in μs and t>=0.

- Graph I vs t for 0<=t<=10 μs (2 μs intervals)
- Find an expression for the total amount of charge that is entering the wire at time t. Q=0C at t=0 μs.
- Graph Q vs t for 0<=t<=10 μs (2 μs intervals).

- See top graph
- Q = (4.0 μC)[1- e-t/(2.0μs)]
- See bottom graph

- I = ei = nevd A
- Define current density as :
J = I/A = nevd (A/m2)

- This quantity is not the same as surface charge density, which implies electrostatic conditions (no moving charge)

The current in wire is doubled. By what factor do the following change?

- Current density
- Conduction-electron density
- Electron drift speed

- J increases by a factor of 2 (J = I/A)
- n remains the same (property of the metal)
- vd increases by a factor of 2 (J = nevd) and e is the charge on the electron, which is constant.