Electric Currents & Resistance
Electric Battery A battery produces electricity by transforming chemical energy into electrical energy. The simplest battery contains two plates or rods made of dissimilar metals (one can be carbon). called electrodes. The electrodes are immersed in a solution, such as dilute acid, called electrolyte. This device is called an electric cell and several cell connected together is a battery.
Electric Current Electric currents are flows of electric charge. The electric current is defined to be the rate at which charges flow across any cross-sectional area. If an amount of charge dq passes through a surface in a time interval dt, then the average current I is given by I = dq dt The SI unit of current is the ampere (A), with 1 A = 1 coulomb/sec. The direction of current is defined to be the direction at which a positive charge would flow through a wire.
Drift Velocity To relate current, a macroscopic quantity, to the microscopic motion of the charges, let s examine a conductor of cross-sectional area A. Let A be the cross-sectional area of a wire and dx be a small slice along the length of the wire. The volume of this small segment of the wire is then dv =A dx (V here is the volume NOT the voltage) Let N be the number of charge carriers contained in this volume and q be the charge per carrier. The number of carriers per unit volume will be n = N dv = N Adx The total charge contained in this volume is thus Although the electron makes a zigzag path through the wire, on average, it continues to move down the electric field at an average velocity called the drift velocity v d : v d = dx # dt " dq = ( nadx)q$ # dq = Nq = ( nadx)q % dq dt I = dq dt = ( naq) dx dt = ( naq)v d = nqav d
Current Density The current per unit cross-sectional area is called current density, J: ( ) J = I A = nqv d A m 2 If the moving charges are negative then the drift velocity is opposite to E. But the current is still in the same direction as E at each point in the conductor. I and J do not depend on the sign of charge, so The general expression s for I and J will be: I = dq dt = n q Av d J = I A = n q v d And the vector current density, is defined as: J = nq v d Note that the current density, J, is a vector, but current is not.
Electric Current A complete circuit is one where current can flow all the way around. By convention, current is defined as flowing from + to Electrons actually flow in the opposite direction.
Ohm s Law To produce an electric current in a circuit, a difference in potential is required. One way of producing a potential difference along a wire is to connect its ends to the opposite terminals of a battery. I V The amount of current in a wire depends not only on the voltage but also on the resistance, R, the wire offers to the flow of electrons. V = IR The unit for resistance is called the ohm and is abbreviated. Resistor Ohmic behavior Non-Ohmic behavior
Ohm s Law In many materials, the current density is linearly dependent on the external electric field E. Their relation is usually expressed as J = E Where, is called the conductivity of the material. It s units are A V m or ( " m )#1 The resistivity of a material is defined as the reciprocal of conductivity. E = J " $ J = I $ # V A V = EL E = V $ L = I A or V = L A I $ L % $ V = L A I " $ # & V = RI L A = R $ % $
Example (1) A 200-km long high-voltage transmission line 2.0 cm in diameter carries a steady current of 1000 A. Assume that the conductor is copper with a free charge density of 8.5 10 28 electrons per cubic meter. (a) How long (in years) does it take one electron to travel the full length of the cable? (b) How long would it take a photon to travel the same distance?
Example (2) A copper wire has a resistance of 10 What will its resistant be if it is only half as long? a) 20 b) 10 c) 5 d) 1 e) None of these
Resistance & Temperature The resistivity of a material varies with temperature T. For metals, the variation is linear over a large range of T: : resistivity at temperature T 0 : resistivity at temperature T 0 : temperature coefficient of resistivity = 0 $% 1+ " ( T # T 0 )& ' If a wire is of constant cross-sectional area A and length L, the change in its resistance due to temperature change will be, R = R 0 #$ 1+ ( T " T 0 )% & R = L A
Example (3) The variation in electrical resistance with temperature can be used to make precise temperature measurements. Platinum is commonly used since it is relatively free from corrosive effects and has a high melting point. Suppose at 20.0 o C the resistance of a platinum resistance thermometer is 164.2. When placed in a particular solution, the resistance is 187.4. What is the temperature of this solution? The temperature coefficient of resistivity for platinum is 3.927 x 10-3 (C o ) -1.
Power & Energy in Electric Circuits Electric energy can be easily transformed into other forms of energy. To find the power (energy rate) transformed by an electric device, recall that the energy transformed when an infinitesimal charge dq moves through a potential difference V is du = V dq Then the power P, which is the rate energy is transformed is: P = U t " = q % # $ t & ' V = IV = IV ab This is precisely the power supplied by the battery. Using "V = IR, one may rewrite the above equation as P = IV = I(IR) = I 2 R P = IV = " # V $ R% & V = V 2 R = V 2 ab R
Example (4) If the electrical energy costs 12 cents per kilowatt-hour, how much does it cost to a) b) burn a 100-W light bulb for 24 hr, and operate an electric oven for 5.0 hr if it carries a current of 20.0 A at 220 V?
The Nature of Sinusoidal Functions Projection of a component (here the y component) of a uniformly rotating vector V0, when plotted against time, generates a sinusoidal functions (sine).
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