L8 - PHY1020U - Winter 2016 PDF

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PHY1020, winter 2016, UOIT Lecture 8, Feb 04

Current and Resistance (chapter 30) - The electron current ie and the current I - The current density J - Ohm's Law

An electric current flows when an electric field is maintained inside a metal. - How does current relate to electric field? - How does current relate to the potential difference across the metal? The Electron Current: - The electrons in a metal are not sitting idle. They undergo random thermal motion The net velocity is however zero. This is static equilibrium. - To move the electrons in one direction we apply an external electric field. Now all free electrons move together (like a liquid in a pipe). - The electrons now move with a drift speed vd. - This controlled motion of electrons is called the current. - The charge carriers in metals are electrons. In semiconductors or solutions they can be different.

The electrons undergo frequent collisions with the nuclei, which affects their drift velocity. If the average time between a collision of the electron and the nuclei is τ, and the electrons mass is m, then the drift velocity is: drift velocity of electrons in field E Question: Is there a dilemma between Newton's second law, and the fact that the electrons move at constant speed vd instead of accelerating in the field E?

The Electron Current ie and current I: The electron current ie is the number of electrons per second that pass through a cross section of a conductor (or wire). [unit s-1] The current I is the rate at which charge moves through a wire. electrical current

where e is the magnitude of the elementary charge (1.6 x 10-19 C). Unit of current is the Ampere : [1 A = 1 C / s]

(Coulomb per second)

Consider a conducting wire of cross section area A that has an electron density n e (number of electrons per unit volume). The wire is subject to an electric field, so electrons will move at drift velocity vd . Consider a section ∆x of the wire, see figure.

- The number of electrons in that section = ne volume = ne ∆x A. - In time ∆t = ∆x / vd, all electrons in length ∆x will have crossed plane C. - The electron current through the wire is hence:

electron current The current is hence: current

NOTE: The direction of the current is actually taken in the opposite direction the electrons move. It is the direction positive charges would move, the same direction of the electric field E. The current density J: A large current I through a thin wire will cause it to melt, while the same I through a wide wire will only cause slight heating. It is hence useful to define the current density J. current density

units of current density is A m-2. where the conductivity σ of a material is: electrical conductivity of the material

units of conductivity is the ohm-1 m -1 or Ω-1 m-1. where 1 Ω = 1 V A-1. Often the resistivity of a material is used:

- The resistivity (and conductivity) depends only on the conducting material, not its shape. - The resistivity in metals increases with temperature, because the atoms vibrate more heavily, increasing the collisions of the electrons.

resistivity of the material

How many electrons are there in metals per unit volume (electron density ne)? Copper, for example, has nearest neighbor distance of 2.56 x 10-10 m and one free electron per atom. So, we expect a density of the order of 1028 electrons/m3.

Example: How much charge and how many electrons flow per second through a wire carrying a current of 1A? By definition 1 A = 1 C / 1 s.

Stop to think: Rank in order, from largest to smallest, the current. a) (2), (4), (3), (1) b) (3), (2), (1), (4) c) (2)=(4), (1)=(3)

Remember: The current is the same through a given wire, even if its cross sectional area changes. However the current density may vary. Stop to think: A current I flows through section a of the wire in the figure. Hence for the currents a) I b > Ia . b) Ib < Ia . c) I b = Ia .

Stop to think: A current I flows through section a of the wire in the figure. Hence for the current densities J a) Jb > Ja b) Jb < Ja c) Jb = Ja

Example : The two wires below are made of the same material. What are the current and the electron drift speed in the 2.0 mm diameter segment of the wire?

A word on Superconductivity In 1911 K. Onnes was studying the conductivity of mercury at very low temperatures. He expected its resistivity to go down with decreasing temperature. What he found instead is a sudden and complete loss of resistance when cooled below 4.2 K. This is great! Charge will continue moving through the 'frictionless' superconductor without an electric field. Many applications: especially where otherwise there would be large dissipation of heat, for example in superconducting electromagnets. Draw-back: 'high temperature' superconductors still need to be cooled to 125 K.

Ohm's law and resistors: A potential difference ∆V applied to a conductor causes a constant electric field inside the wire.

As a result, a current I flows.

Ohm's law which is Ohm's law. It states a linear relation between current and potential difference.

resistance of a conducting wire

where σ is the conductivity and ρ is the resistivity. R depends both on the material (ρ) and shape (length L, area A). [units of R is the ohm (Ω): 1 Ω = 1 V A-1].

various shapes of resistors (conductors) Note: not all materials obey Ohm's law

Example: In an electric circuit, a 100 ohm resistor is connected through two 1.5 m long, 1.0 mm in diameter copper wires to a battery [resistivity of copper = 1.7 x 10-8 ohm.m]. - What is the resistance of each copper wire? Compare it to that of the resistor. - If the battery output is 10 V, what is the current through the resistor?

Example: Consider a nichrome wire, 3.0 m long and with a diameter of 2.0 mm, connected to an ideal 1.5 V battery. Find the following: a) electric field in the wire. b) resistance of the wire. Nichrome's resistivity is 1.5 x 10-6 Ω .m c) current through the wire. d) current density in the wire. e) drift velocity of the electrons in the wire. Assume the electron density in nichrome is 8.0 × 1028 electrons m-3. f) time between collisions....