Electric Current and Direct Current Circuits

2 Why bother with Electric Potential V, Capacitance, and Electric Fields? The electric potential difference between point a) and point b) in a circuit is the force driving current from a) to b) Capacitance is universal,  Capacitance limits the speed of switching circuits  Capacitance stores energy Electric Field energy is the energy of waves, light.

3 Whenever there is a net movement of charge, there exists an electrical current. A current can flow in a wire: usually electrons. A current can flow in a liquid solution: For example Na+, and K+ ions across a nerve cell membrane. A current can flow in air or free space: electron or ion beam, lightning. Electric Current

4 Unit of measure of Electric Current If a charge  Q moves through a “surface” A in a time  t, then there is a current I: The unit of current is the Ampere (A): 1 A= 1 Coulomb/sec. By convention, the direction of the current is the direction of flow of the positive charges. If electrons flow to the left, that is a positive current to the right. ee I

5 Ohm’s Law For many materials, the current I is directly proportional to the voltage difference V. We define the resistance, R, of such a material to be: The unit of resistance is Ohms  : 1  Volt/Amp Common resistors used in electrical circuits range from a few  to  (10 6 W). If R is constant: doesn’t depend on current, or history of current flow, and only small variation with temperature, atmospheric pressure, etc, the material is said to be ohmic, and we write Ohm’s Law:

6 Fluid Analogy of Resistance A fluid (liquid or gas) will not flow through a narrow tube unless there is a pressure difference between the input and output ends. The pressure difference can be provided by external pressure, or by gravity. The longer the tube, or the narrower the tube, the larger a pressure difference (or gravity gradient) is required to maintain the same flow. v P in P out Fluid flow Liquid flow

7 An object which provides resistance to current flow is called a resistor. The actual resistance depends on: properties of the material (resistivity) the geometry (length and cross sectional area) For a conductor of length L and cross-sectional area A, the resistance is R=  L/A, where  is called the resistivity. Resistivity  L Area A

8 Wires & Resistors in Circuits A piece of wire is a resistor. However, for good conductors like Cu, Al, Au, Ag, the resistivity is extremely low. When we analyze a circuit containing wires and other elements (such as light bulbs), the resistance of the wires is so low that we can [usually] pretend the wires are perfect conductors. Current can flow in the wire even though the potential is everywhere the same inside each separate piece of wire.

9 Sample Resistivity values Material Resistivity  (  m) Silver (Ag) 1.59   Tungsten (W) 5.6   Nichrome 100   Graphite ( C ) 3500   Si 2.5   Glass> 10 10

10 Walker problem #8 A silver wire is 4.5 m long and 0.40 mm in diameter. What is its resistance? 

11 A typical cell membrane is 8.0 nm thick and has an electrical resistivity of 1.3x10 7  ·m. If the potential difference between the inner and outer surfaces of a cell membrane is 80 mV, how much current flows through a square area of membrane 1.3 µm on a side? Walker problem #14

12 Resistors in Circuits In drawing a circuit, the symbol for a resistor is This zigzag pattern is a visual reminder that the material of the resistor impedes the flow of charge, and it requires a potential difference V between the two ends to drive current through the resistor. Current flows from higher value of potential to lower value of potential

13 Simple Battery Circuit A battery is like a pump  A pump raises fluid by a height h.  A battery pumps charge up to a higher potential. V +  0 R I I I = V/R Current is the same everywhere. Voltage varies from point to point around loop.

14 An Incandescent Light Bulb is a Resistor (but R depends on Temperature T of filament, and T depends on current I).

15 Question: Which Circuit will light the bulb? A)B)C)

16 Recall that resistance is like an internal friction - energy is dissipated. The amount of energy dissipated when a charge  Q flows down a voltage drop V in a time  t is the power P: P =  U/  t =(  Q·V/  t) = IV SI unit: watt, W= Amp·Volt = C V/s = J/s For a resistor, P=IV can be rewritten with Ohm’s Law V=IR, P = I 2 R = V 2 /R Power is not Energy, Power is rate of consumption (or production) of energy Large power plants produce between 100 MW and 1GW of power. This power is then dissipated in the resistors and other dissipative circuits in our electronic appliances, in the resistance of the windings of electric motors, or is used to charge batteries for later use. Power in Electric Circuits

17 Energy and Power Energy Usage: Power times time = Energy consumed 1 kilowatt-hour = (1000 W)(3600 s) = (1000 J/s)(3600 s) = 3.6  10 6 J Electricity in VA costs about $0.15 per KW  hr A typical household uses 1KW of power, on average. There are ~8800 hours in a year In one year, each household consumes 8800 KW  hr, (8800 KW·hr)($0.15) = $1320.

18 Direct Current (DC) Circuits A circuit is a loop comprised of elements such as batteries, wires, resistors, and capacitors through which current flows. Current can only flow around a loop if the loop is continuous. Any break in the loop must be described by the capacitance of the gap, which allows charge to build up as current flows onto the capacitors. For current to continue flowing in a circuit with non-zero resistance, there must be an energy source. This source is often a battery. A battery provides a voltage difference across its terminals.

19 Simple Battery Circuit An incandescent light bulb can be approximated as an ideal resistor (this is a bad approximation, because most light bulbs have a very strong temperature dependence to the resistance). V=IR 5 Watt bulb with 3 V battery: P= V 2 / R R = V 2 /P = (3V) 2 /(5 A·V) R = 1.8 V/A = 1.8 .

20 Direct Current (DC) Circuits - MORE Includes:batteries, resistors, capacitors Kirchoff’s Rules- conservation of charge (Laws) follow from: ( junction rule, valid at any junction) - conservation of energy (loop rule, valid for any loop) With emf (  ):constant current can be maintained “charge pump” forces electrons to move in a direction opposite to the electric field SI unit for emfVolt (V) No resistance connecting wires of the loop

21 Kirchhoff’s Rules Any charge must move around any closed loop with emf Any charge must gain as much energy as it loses Loss: IR – potential drop across resistor Gain: chemical energy from the battery (charge go reverse direction from  ) Often what seems to be a complicated circuit can be reduced to a simple one, but not always. For more complicated circuits we must apply Kirchhoff’s Rules: Junction Rule: The sum of currents entering a junction equals the sum of currents leaving a junction. Loop Rule: The sum of the potential difference across all the elements around any closed circuit loop must be zero.

22 Combining Circuit Elements Any two circuit elements can be combined in two different ways: in series - with one right after the other, or in parallel - with one right next to the other. SeriesParallel Combination Combination

23 Resistors in Series By the conservation of charge,  The same current I flows through all three resistors, and through the battery.  Junction rule at a, b, c, d, separately a b c d Ohm’s law: + E – IR1 – IR2 – IR3 = 0 E = IR1 + IR2 + IR3 IR total = I(R1+R2+R3) R total = (R1+R2+R3)

24 Resistors in Parallel The current I splits into three (non-equal) branches such that I=I 1 +I 2 +I 3. Ohm’s law:  = (  /R 1 +  /R 2 +  /R 3 ) R eq =  R eq (  /R 1 +  /R 2 +  /R 3 )  / R eq = (  /R 1 +  /R 2 +  /R 3 )

25 Resistors in both Series and Parallel Combine the first two in parallel to obtain equivalent resistance R/2. Combine three in series to obtain equivalent resistance R + (R/2) + R = 2.5 R.

26 What is the equivalent resistance of circuit? 1.0  2.0  1.0  3 V

27 What is the current in each resistor? 1.0  2.0  1.0  3 V

 2.0  1.0  3 V What is the power dissipated in each resistor?

29 Capacitors are used in electronic circuits. The symbol for a capacitor is We can also combine separate capacitors into one effective or equivalent capacitor. 2 capacitors can be combined either in parallel or in series. Series Parallel Combination Combination C 1 C 2 C2C2C2C2 Circuits containing Capacitors

30 Parallel vs. Series Combination ParallelSeries charge Q 1, Q 2 charge on each is Q total Q=Q 1 + Q 2 total charge is Q voltage on each is V voltage V 1 + V 2 = V Q 1 =C 1 V Q=C 1 V 1 Q 2 =C 2 V Q=C 2 V 2 Q=C eff V Q= C eff V = C eff (V1+V2) C eff =C 1 +C 2 1/C eff =1/C 1 +1/C 2 C 1 C 2 Q QQ Q QQ

31 RC Circuits We can construct circuits with more than just resistor, for example, a resistor, a capacitor, and a switch: When the switch is closed the current will not remain constant. Capacitor acts as an open circuit: I=0 in branch with capacitor under study state condition.

32 Capacitor Charging Lets assume that at time t=0, the capacitor is uncharged, and we close the switch. We can show that the charge on the capacitor at some later time t is: q=q max (1-e -t/RC ) RC is known as the time constant , and q max is the maximum amount of charge that the capacitor will acquire: q max =C 

33 Capacitor Discharging Consider this circuit with the capacitor fully charged at time t=0: It can be shown that the charge on the capacitor is given by: q=q max e -t/RC