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Lecture 10 Ch26 - PH 222-3A Spring 2007 CURRENT AND...

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Unformatted text preview: PH 222-3A Spring 2007 CURRENT AND RESISTANCE Lecture 10 L Chapter 26 (Halliday/Resnick/Walker, Fundamentals of Physics 8th edition) 1 Chapter 26 Current and Resistance In this chapter we will introduce the following new concepts: p g p -Electric current ( symbol i ) -Electric current density vector (symbol J ) -Drift speed (symbol vd ) -Resistance (symbol R ) and resistivity (symbol ) of a conductor Ohmic non-Ohmic -Ohmic and non Ohmic conductors We will also cover the following topics: -Ohm's law -Power in electric circuits Electric Current Consider the conductor shown in fig. a. All the points inside the conductor and on its surface are at the same potential. The free electrons inside the conductor move in random directions and thus there is no net charge transport. A B We now make a break in the conductor and insert a battery as shown in fig. b. Points A and B are now at potentials VA and VB , respectively d ti l ( VA - VB = V , the voltage of the battery). The situation is not static any more, but charges move inside the conductor so that there is a net charge flow in a particular direction. direction We define this net flow of electric charge as "electric current." Consider the conductor shown in the figure. It is connected to a battery (not shown) and thus charges move through the conductor. h h h h h d Consider one of the cross sections through the conductor ( aa or bb or cc ). The electric current i is defined as i = dq i= dt conductor +q dq . dt Current = rate at which charge flows Current SI Unit: C/s, known as the "ampere" v Current Direction An electric current is represented by an arrow, which has the same direction as the charge velocity. The sense of the current arro c rrent arrow is defined as follows: follo s: 1. If the current is due to the motion of positive charges, the current arrow is parallel to the charge velocity v . 2. If the current is due to the motion of negative charges, the current arrow is antiparallel to the charge velocity v . i conductor d t -q i v conductor +q i conductor -q i v A J Current Density Current density is a vector that is defined as follows: i SI unit for J : A / m2 A The direction of J is the same as that of the current. The current through a conductor of cross-sectional area A is given by the equation i = JA if the current density is constant. y Its magnitude is J = If J is not constant, then i = J dA. v A J i J= A We note that even though the current density is a vector the electric current is not. This is illustrated in the figure to the left. An incoming current i0 branches at point a into two currents, i1 and i2 . Current i0 = i1 + i2 . This equation expresses the conservation of charge at point a. Please note that we have not used vector addition. Drift Speed When a current flows through a conductor g the electric field causes the charges to move with a constant drift speed vd . This drift speed is i superimposed on the random motion of th i d th d ti f the charges. J = nvd e J = nevd Consider the conductor of cross-sectional area A shown in the figure. We assume p g g that the current in the conductor consists of positive charges. The total charge q within a length L is given by q = ( nAL ) e, where n is a number of carriers per unit volume. This charge moves through area A L q nALe AL in a time t = . The current is i = = = nAvd e. vd t L / vd The current density is J = In vector form: J = nevd . i nAvd e = = nvd e. A A i + V V i - Resistance If we apply a voltage V across a conductor (see figure) a current i will flow through the conductor. V We define the conductor resistance as the ratio R = . i V SI Unit for R : = the ohm (symbol ) A A conductor across which we apply a voltage V = 1 volt and results in a current i = 1 ampere is defined as having resistance of 1 . Q : Why not use the symbol "O" instead of " "? A : Suppose we had a 1000 resistor. , y We would then write: 1000 O, which can easily R= R be mistakenly read as 10000 . A conductor whose function is to provide a specified resistance is known as a "resistor." ifi d i t i k " it " The symbol is given to the left. i Resistivity E + E = J V J =E Unlike the electrostatic case, the electric field in the conductor of the figure is not zero. We d fi as d t f th fi i t W define resistivity of the conductor the ratio = In vector form: E = J . E . J SI unit for : V/m V = m = m 2 A/m A 1 E R= L A The conductivity is defined as = . Using U i , the previous equation takes the form: J = E. th i ti t k th f Consider the conductor shown in the figure above. The electric field inside the V i conductor is E = . The current density is J = . We substitute E and J into L A A L E V /L V A = =R R= . equation = and get: = J i/ A i L L A - 0 = 0 (T - T0 ) Variation of Resistivity with Temperature In the figure we plot the resistivity of copper as a function of temperature T . The dependence of on T is almost linear. Similar dependence is observed in many conductors. The f ll i Th following empirical equation is used f many practical applications: ii l ti i d for ti l li ti - 0 = 0 ( T - T0 ) . The constant is known as the "temperature coefficient of resistivity." The constant T0 is a reference temperature p y p usually taken to be room temperature (T0 = 293 K ), and 0 is the resistivity at T0 . For copper, o = 1.69 10-8 m. Note : Temperature enters the equation above as a difference ( T - T0 ) . Thus either the Celsius or the Kelvin temperature scale can be used. Ohm's Law. A resistor was defined as a conductor whose resistance does not change with the voltage V applied across it. In fi b we plot the current i through a resistor ih h l li d i fig. l h h h i as a function of V . The plot (known as the "i - V curve" ) is a straight line that passes through the origin. Such a conductor is said to be "Ohmic" and it obeys Ohm's law, which states: The current i through a conductor is proportional to the voltage V applied across it. Not all conductors obey Ohm's law (these are known as "non - Ohmic"). An example is given in fig. c where we plot i versus V for a semiconductor diode. The ratio fig diode V / i (and thus the resistance R ) is not constant. As a matter of fact, the diode does not conduct for negative voltage values. Note : Ohm's "law" is in reality a definition of Ohmic conductors (defined as the conductors that obey Ohm's law). A Microscopic View of Ohm's Law In order to understand why some materials such as metals obey Ohm's law, we must look into the details obe la e m st of the conduction process at the atomic level. vd F A schematic of an Ohmic conductor such as copper is shown in the figure. We assume that there are free electrons that move around in random directions with an effective speed veff = 1.6 106 m/s. The free 16 m/s electrons suffer collisions with the stationary copper atoms. A schematic of a free electron path is shown in the figure using the dashed gray line. The electron starts at point A and ends at point B. We now assume that an electric field E is applied. The new electron path is indicated by the dashed green line. Under the action of the electric force the electron acquires a small drift speed vd . The electron drifts to the right and ends at point B. Consider the motion of one of the free electrons. We assume that the average time between collisions with th copper atoms is equal to . The electric field ith the t i lt Th l t i fi ld exerts a force F = eE on the electron, resulting in an F eE = . The drift speed is given vd m m eE F by the equation vd = a = (eq. 1). m J We can also get vd from the equation J = nevd vd = (eq. 2). ne J eE m If we compare equations 1 and 2 we get: vd = = E = 2 J. ne m ne If we compare the last equation with E = J we conclude that m = 2 . This is a statement of Ohm's law (the resistance of the ne conductor does not depend on voltage and thus E ). This is because m, n, and e are constants. The time can also be considered to be independent of E since the drift speed vd is so much smaller than veff . acceleration a = P = iV Power in Electric Circuits Consider the circuit shown in the figure. A battery of voltage V is connected across the terminals a and b of a device. This can be a resistor, a motor, etc. The battery maintains a potential difference V between the terminals a and b and thus a current i flows in the circuit as shown in the figure. During the time interval dt a charge dq = idt moves between the terminals. W note that Va >Vb . b h i l We h The potential energy of the charge decreases by an amount dU = Vdq = Vidt. Using energy conservation we conclude that the lost energy has been transferred by the battery to the device and has been converted into some other form of energy. y y gy The rate at which energy is transferred to the device is known as "power" and it is dU Vidt = = Vi. dt dt SI unit for P : V A It is known as the "watt" (symbol W). equal to P = V P=i R 2 V2 P= R If the device connected to the battery is a resistor R then the energy transferred by the battery is converted as heat that appears on R. If we combine the equation P = iV V with Ohm's law i = , we get the following two equivalent expressions for R the rate at which heat is dissipated on R: P=i R 2 and V2 P= R ...
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