Electric Current and DC Circuits Ch. 25 Electric Current and DC Circuits

Chapter Overview Definition of Current Ohm’s Law Resistance – Conduction in Metals Kirchhoff’s Laws Analysis of DC Circuits RC Circuits

Current Up to this point we have been concerned with charges that don’t move – Static When charges do move, then an electric current flows Current, usually denoted by the letter i is that rate at which charge moves. In other words how much charges flows past a point per time

Current i = current q = charge t = time SI Units – ampere Symbol A Fundamental Unit (More on this later)

The ampere is a fundamental unit, so the coulomb is a derived unit The ampere is a fundamental unit, so the coulomb is a derived unit. Express the coulomb in terms of fundamental units A/s A·s s/A None of he above 1 2 3 4 5

Batteries are often rated in amp·hours Batteries are often rated in amp·hours. What type of quantity does an amp∙hour represent? Current charge Electric Potential Capacitance None of the above 1 2 3 4 5

Ex. How many coulombs of charge are stored in 60 A·hr battery?

Solution i=Δq/Δt Δq=iΔt Δq=60 A x 1 hr =60A x 1 hr x 3600 s/1 hr =216000 C

Ex. 10000 protons fIow through a detector every. 05 s Ex. 10000 protons fIow through a detector every .05 s. What is the current flowing through the detector?

Soln. Current i=Δq/Δt i=10000x1.602x10-19 C/.05 s i= 3.2 x 10-14 A Protons Detector

Microscopic View of Current At the microscopic level, current is made by individual charges moving at a speed vd in the material A charge will travel the distance x in a time given by t = x/vd x vd q

Microscopic view of current The total amount of charge that flows through the gray shaded volume in time Δt is ΔQ = nqV where n is the number of charges per volume, V is the volume of the gray shaded area, and q is the charge of an individual charge V = xA = vdΔtA So ΔQ = nqvdAΔt

Drift Velocity I = ΔQ/Δt = nqAvd vd is the drift velocity. It represents the average speed of charges in conductor Ex. A copper wire has a radius of .50 mm. It carries a current of .25 A. What is the drift velocity of the electrons in the wire. Assume 1 free electron per atom. (ρcu = 8.93 g /cm3)

There will be Avagadro’s number of charges in 1 mole of copper There will be Avagadro’s number of charges in 1 mole of copper. the molar mass of copper is 63.5 g. The volume of 1 mole of copper is V = 63.5 g/8.93 g/cm3 = 7.31 cm3 = 7.31 x 10-6 m3

n = NA / Vmol = 6.022 x 1023/7.31 x 10-6 m3 = 8.24 x 1028 electrons /m3 vd = i/nqA = .25 A/(8.24 x 1028/m3 x 1.602 x 10-19 As x 3.14 x(5 x 10-4 m)2) = 2.4 x 10-5 m/s

Ohm’s Law What factors determine how much current will flow in a circuit (BRST)

Ohm’s Law Potential Difference and the material properties determine current that flows in a circuit

Ohm’s Law Current is proportional to potential difference “Resistance” limits the amount of current that flows Experimental Relationship found by Georg Simon Ohm Not always true – e.g. diodes, transistors,…

Resistance Units – ohm denoted by Ω What is the ohm in terms of V and A What is the ohm in terms of fundamental units

Units Ω=V/A Ω = kg m2 /(A2 s3 )

Resistivity What factors determine the resistance of a piece of metal? (BRST)

Resistivity Length – L Cross sectional Area – A resistivity ρ – material property (see table in text p. 792)

Two different metals with different resistivities have the same length Two different metals with different resistivities have the same length. Which metal will have the higher resistance? The metal with the greater ρ The metal with the smaller ρ The resistances will be the same Cannot be determined 1 2 3 4 5

Temperature Dependence of Resistance If you measure the resistance of a light bulb cold and then measure it when it is glowing, do you get the same resistance? (BRST)

Temperature Dependence of Resistance Resistance increases with increasing temperature for metals It decreases with increasing temperature for semiconductors For conductors R = R20(1 + α(T- 20 C°)) α is the Temperature coefficient of resistance (see p. 792)

Ex. Find the temperature of the filament of a light bulb (assume W) by measuring the resistance when cold and glowing.

Power When we apply a potential difference across a resistor, it gets hot What determines the power given off by a resistor?

Power Work = qΔV Power = Work/time P = qΔV/t but q/t = i P = iΔV

Power Combine the expression for electric power P = iΔV with Ohm’s law ΔV = iR P = iΔV = i2R = (ΔV )2/R

Ex. A 100 W bulb is designed to emit 100 W when connected to a 120 V circuit. a) Draw a sketch and a schematic. b) What is the resistance of the bulb and the current drawn when connected to a 120 V outlet? c) Assuming the resistance doesn’t change, what would be the power output if the bulb was connected to a 240 V circuit?

The electric bill Power is a rate – it tells you how much energy per time is being used

The electric company bills you in units of kw∙hr. What is a kw∙hr The electric company bills you in units of kw∙hr. What is a kw∙hr? (TPS) It is a unit of power It is a unit of energy It is a unit of current It is a unit of potential difference Not enough information given 1 2 3 4 5

The electric company bills you in units of kw∙hr. What is a kw∙hr The electric company bills you in units of kw∙hr. What is a kw∙hr? (TPS) It is a unit of power It is a unit of energy It is a unit of current It is a unit of potential difference Not enough information given 1 2 3 4 5

EX. How many joules are in a kW∙hr?

For the circuit shown below how does the current flowing through A compare to that flowing through B It is the same It is greater at A It is greater at B It cannot be determined 3.0 Ω B A V 6.0 Ω 1 2 3 4 5

The current is twice as large The current is ½ as large You connect two identical resistors in series across a 6.0 V battery. How does the current in the circuit compare to that when a single resistor is connected across the battery There is no difference The current is twice as large The current is ½ as large Cannot be determined 1 2 3 4 5

Resistors in Series When we add resistors is series, the current decreases since the resistance increases We define an equivalent resistance as a single resistor which produces the same current when attached to the same potential as the combination of resistors

Series equivalent

Equivalent Series Resistance We want i to be the same V= iReq V1 = iR1, V2 = iR2, V3 = iR3 V = V1 + V2 + V3 = iR1 + iR2 + iR3 iR1 + iR2 + iR3 = i(R1 + R2 + R3) = iReq So Req = R1 + R2 + R3

Equivalent Series Resistance How would this result change if there were four resistors in series? In general as more resistors are added in series, the resistance increases so the current decreases

Ex. a) Find the equivalent resistance for the following circuit Ex. a) Find the equivalent resistance for the following circuit. b) Find the current In the circuit c) Find the potential drop across each resistor d) Find the power dissipated by each resistor. e) Find the power Supplied by the power supply.

The current is twice as large The current is ½ as large You connect two identical resistors in parallel across a 6.0 V battery. How does the current supplied by the battery compare to that when a single resistor is connected across the battery? There is no difference The current is twice as large The current is ½ as large Cannot be determined 1 2 3 4 5

Resistors in Parallel When we add resistors is parallel, the current increases The effective resistance must then decrease How can that be? (BRST)

Resistors in Parallel There are more branches for current to follow in a parallel circuit, so current can be larger

Resistors in Parallel What is the same for the three resistors shown? (GR)

Resistors in Parallel The potential difference across each resistor is the same Define i1 = V/R1, i2 = V/R2, i3 = V/R3 How do the currents combine?

Parallel Equivalent Resistance We define the equivalent resistance as a single resistor that will draw the same current from the power supply

Parallel Equivalent resistance The currents add. Why? i = i1 + i2 + i3 = V/R1 + V/R2 + V/R3 = V/Req

a) Find the equivalent resistance a) Find the equivalent resistance. b) Find the current flowing through each resistor. c) Find the current supplied by the power supply

Find the equivalent resistance for the following network

Kirchhoff’s laws The rules for series and parallel resistances are examples of Kirchhoff’s Laws Voltage Law - Sum of the potential differences around a closed loop is 0 Current Law- the sum of currents at a node is 0

Voltage Law A loop is a closed path in a circuit

Current law A node is a point in a circuit where several wires join

For the circuit shown below a) choose currents for each branch of the circuit. b) For the choice of currents you’ve made, label the higher potential side of each resistor with a + and the lower potent side with a – c) Use Kirchoff ‘s laws to write a closed system of equations for the currents. d) Solve for the currents. e) Find the potential difference across each resistor. f) Find the power dissipated by each resistor. g) Find the power supplied by each’ power supply

RC Circuits A resistor in series with a capacitor makes an RC circuit RC circuits have many applications – e.g. camera flashes

RC Circuits - Charging Use Kirchhoff’s Voltage Law to analyze the circuit shown below R V C

RC Circuits V – VR – VC = 0 V – iR – Q/C = 0 But i = dQ/dt V - R dQ/dt – 1/C Q = 0

RC Circuits Kirchhoff’s Voltage Law gives a differential equation for the charge dQ/dt = V/R - Q/RC Assuming the capacitor is initially discharged the solution (You’ll work it out in lab) Q = CV(1 – e-t/(RC))

Charging a Capacitor When t = RC = 5 s, Vc = .63 * 6V =3.78 V

Current in a Charging RC Circuit i = dq/dt = V/R e-t/RC The current exponentially decays with the same time constant

Discharging a Capacitor Assume that the capacitor is initially charged with Q0 = CV0 What will happen when the switch is closed R C

Discharging a Capacitor -R dq/dt – q/C = 0 dq/dt = -q/(RC) A solution is q = CV0 e-t/(RC) Solution is exponential decay RC is the time constant

Ex. a) What fraction of the original charge remains when t = RC Ex. a) What fraction of the original charge remains when t = RC? b) At what time is charge reduced to a fraction f of the initial amount?

Current in a Discharging RC Circuit i = dq/dt = -V/R e-t/RC The current exponentially decays with the same time constant It flows in the opposite direction