Ohm’s Law and Resistance. Resistivity. Today’s agenda: Electric Current. You must know the definition of current, and be able to use it in solving problems. Current Density. You must understand the difference between current and current density, and be able to use current density in solving problems. Ohm’s Law and Resistance. You must be able to use Ohm’s Law and electrical resistance in solving circuit problems. Resistivity. You must understand the relationship between resistance and resistivity, and be able to use calculate resistivity and associated quantities. Temperature Dependence of Resistivity. You must be able to use the temperature coefficient of resistivity to solve problems involving changing temperatures.
Current density: charge per area per time (current / area). When we study details of charge transport, we use the concept of current density. Current density is the amount of charge that flows across a unit of area in a unit of time. + + + + Current density: charge per area per time (current / area).
Simpler, less-general OSE on next page. A current density J flowing through an infinitesimal area dA produces an infinitesimal current dI. dA J Current density is a vector. Its direction is the direction of the velocity of positive charge carriers. The total current passing through A is just No OSE’s on this page. Simpler, less-general OSE on next page. Current density: charge per area per time.
A J If J is constant and parallel to dA (like in a wire), then Now let’s take a “microscopic” view of current and calculate J. If n is the number of charges per volume, then the number of charges that pass through a surface A in a time t is vt v A q
The total amount of charge passing through A is the number of charges times the charge of each. vt v A q Divide by t to get the current… …and by A to get J:
To account for the vector nature of the current density, Not quite “official” yet. and if the charge carriers are electrons, q=-e so that Not quite “official” yet. The – sign demonstrates that the velocity of the electrons is antiparallel to the conventional current direction.
Thanks to Dr. Yew San Hor for this slide. Currents in Materials Metals are conductors because they have “free” electrons, which are not bound to metal atoms. In a cubic meter of a typical conductor there roughly 1028 free electrons, moving with typical speeds of 1,000,000 m/s… …but the electrons move in random directions, and there is no net flow of charge, until you apply an electric field. Thanks to Dr. Yew San Hor for this slide.
A greatly oversimplified model, but the “idea” is useful. electron “drift” velocity - just one electron shown, for simplicity inside a conductor The electric field accelerates the electron, but only until the electron collides with a “scattering center.” Then the electron’s velocity is randomized and the acceleration begins again. Some predictions based on this model are off by a factor or 10 or so, but with the inclusion of some quantum mechanics it becomes accurate. The “scattering” idea is useful. A greatly oversimplified model, but the “idea” is useful.
In particular, the velocity that should be used in Even though some details of the model on the previous slide are wrong, it points us in the right direction, and works when you take quantum mechanics into account. In particular, the velocity that should be used in is not the charge carrier’s velocity (electrons in this example). Instead, we should the use net velocity of the collection of electrons, the net velocity caused by the electric field. Quantum mechanics shows us how to deal correctly with the collection of electrons. This “net velocity” is like the terminal velocity of a parachutist; we call it the “drift velocity.”
It’s the drift velocity that we should use in our equations for current and current density in conductors:
Example: the 12-gauge copper wire in a home has a cross-sectional area of 3.31x10-6 m2 and carries a current of 10 A. The conduction electron density in copper is 8.49x1028 electrons/m3. Calculate the drift speed of the electrons.