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Electrical Properties of Materials I. Current – I A. Definition: Current is defined as the rate at which charge flows through a surface: + + B. Direction: The direction of positive current flow is chosen as the direction that _____________________________ charge flows. (regardless of the true nature of the charge carrier) EXAMPLE 1: I + + EXAMPLE 2: I - C. Units : II. Model for Current in a Conductor A. Although the macroscopic view of current is useful in building and troubleshooting electronic circuits, the actual flow of charge is a microscopic process involving the motion of electrons, ions, or other microscopic charge carriers. Thus, it is not surprising that the electrical properties of a material are strongly dependent on the atomic makeup of a material and requires “small physics” (Quantum Mechanics and Solid State) to completely describe. B. A microscopic understanding of any material used in engineering is important since it allows us to predict the physical, chemical, and electrical properties of the material and provides a means for modifying and improving these properties. This field of engineering is called Material Science. Many of the advances in electronics, civil engineering, chemical engineering, etc. are due to the availability of new materials with improved properties. Only physicists build objects using physics strings and pulleys to understand the laws of nature. Engineers build actual devices that require real materials!!! C. The flow of current through a conductor is the simplest case of current flow and the only type we consider in this course. The outer electrons in a good conductor (ex. a metal) are very weakly bound to the atom. Thus, we can consider them to be free like atoms in a gas. This removes our need for the more exact and mathematically complex physics of Quantum Mechanics. “Random Motion in Ideal Gas” “Random Motion of Charges in a Conductor” Push Push + - Voltage “Random Motion + Gas Flows to Right” D. “Random Motion + Current Flow to Right” Microscopic View n ≡ number of charge carriers per volume Let Vd ≡ average net velocity of charge carrier (drift velocity) q ≡ charge on a carrier A ≡ cross sectional area Thus, we have dQ = (number of carriers) (charge per carrier) dQ = but dx = dQ = I We define the current density as J ≡ . A Thus, we have for the current density the result: J =nqVd At the microscopic level, current flow in a material can occur in multiple dimensions and may have different drift velocities in each direction. Thus, the current density can be written in full vector form as: J = n q Vd EXAMPLE: What is the magnitude of the drift velocity of an electron in aluminum, if 5.00 A of current flow through an aluminum wire of area 4.00x10-6 m2. (Assume that each aluminum atom provides one electron for current conduction) I A GIVEN: A = 4.00x10-6 m2, ρ = 2.70 g/cm3, I = 5.00 A, and q = 1.60x10-19 C SOLUTION: III. Ohm’s Law George Simon Ohm showed that for some materials the current that flows through a material is _____________________ ____________________________ to the potential difference across the material. Such materials are called ___________________ materials. V A Test Material - + Supply Voltage Voltage Current “__________________ Material” Current “ __________________ Material” NOTE: Ohm’s Law is not a law!!! Most materials do not have the correct I-V curve. IV. Resistance and Resistors A. Resistance is a measure of a materials _____________________ to the flow of charge through the material. It is the ______________________ of our I-V graph. For a ____________________ material of fixed dimension, the resistance is constant for a given temperature. B. The unit of resistance is the ________________. Its symbol is the ____________. C. Resistor – An electrical element that transfers electrical energy into heat or mechanical energy while obeying Ohm’s Law. Resistor Symbol: Ohm’s Law: EXAMPLE 1: What is the resistance of a material if you measure a voltage drop of 10.0 V when 4.00 mA of current flows through it? EXAMPLE 2: What is the voltage drop across the 6.00 kΩ resistor in the circuit below? 2.00 kΩ 4.00 mA + 6.00 kΩ V - V. Resistivity and Conductivity A. The resistance of an element depends not only on the material from which the element was constructed but also upon the: 1) ________________________________________________ 2) ________________________________________________ B. The resistivity, ρ, is an intrinsic property of material at a given temperature just like density, melting point, etc. C. The relationship that connects the intrinsic material property of resistivity and a resistor’s resistance is: D. Conductivity, σ, is defined as the inverse of resistivity. E. Ohm’s Law for a conductor can be written in terms of the resistivity as EXAMPLE: You are required to construct a 2.00 Ω resistor using a carbon rod with a cross sectional area of 1.00x10-2 cm2 piece of carbon. How long must your carbon rod be? L A GIVEN: r = 3.5x10-5 W • m “Table in book” SOLUTION: VI. Temperature Effects of Resistance and Resistivity A. As temperature increases, the resistance and resistivity of a material ______________________________________________________!! Increasing the temperature of a material causes the bound atoms to vibrate at increased speeds. These atoms scatter the moving charge carriers thereby reducing current flow. (Consider people trying to travel on a moving sidewalk without hitting sliding doors that are constantly opening and closing. Higher temperature corresponds to faster doors!!) B. Mathematical Approximation Technique MATH FACT: Any analytical function, f(t), can be represented as a power series in t such that ∞ 𝑓(𝑡) = ∑ 𝐴𝑛 (𝑡 − 𝑡𝑜 )𝑛 = 𝐴0 + 𝐴1 (𝑡 − 𝑡𝑜 ) + 𝐴2 (𝑡 − 𝑡𝑜 )2 + ⋯ 𝑛=0 where the expansion coefficients are found by 𝐴𝑛 = 𝑑𝑛 𝑓 | 𝑑𝑡 𝑛 𝑡=𝑡 𝑜 𝑛! By analytic, we mean that the function is a continuous, single valued finite function. This requirement is met by most functions in physics and engineering and provides us with a powerful tool for modeling real world problems. The problem of handling non-analytic functions using the Laurent Series will be covered in your math course work. By wisely choosing a value t0 such that t-t0 is small, it is possible to approximate the function with a power series involving only a few terms!! C. Temperature Model for Resistivitivy The resistivity of a material is an analytical function of temperature. Using the initial temperature of the material for which we know its resistivity as t0, we can expand the resistivity in terms of a power series as follows: 𝜌(𝑡) = Using the fact that (t – t0) = ∆t and limiting ourselves to small temperature changes, we have that: EXAMPLE: What is the resistivity of silver at 100 C°? SOLUTION: From tables in the book, we have for silver at 20 C° a resistivity of 1.47x10-8 Ω • m and a temperature coefficient of resistivity of 3.8x10-3 (C°)-1. VII. Power in Electric Circuits A. The power supplied or consumed by any electrical circuit element (resistor, Capacitor, battery, inductor, etc.) is given by B. Special Formula for Resistor Since resistors obey Ohm’s law, we can rewrite our general formula as EXAMPLE: What is the power being dissipated by the resistor and stored by the capacitor in the circuit below: 5.00 Ω 6.00 A + C 4.00 V -