Current and Resistance

Whenever there is a net movement of charge, there exists an electrical current. If a charge Q moves perpendicularly through a “surface” of area A in a time t, then there is a current I:

The unit of current is the Ampere (A): 1 A = 1 C/s.

By convention, the direction of the current is the direction of the flow of positive charges. The actual charge carriers are electrons; hence they move in the opposite direction to I.

t

QI

Chapter 21Electric Current and Direct-Current Circuits

Batteries and Electromotive Force (emf)

Any device which increases the potential energy of charges which flow through it is called a source of emf,

SI unit for emf : Volt (V)

The emf may originate from a chemical reaction as in a battery or from mechanical motion such as in a generator.

A battery is a device that uses chemical reactions to produce a potential difference between its two terminals.

Water flow as analogy for electric current

Resistance and Ohm’s Law

In order for a current I to flow there must be a potential difference, or voltage V, across the conducting material. We define the resistance, R, of a material to be:

The unit of resistance is Ohms : 1 V/A

For many materials, R is constant (independent of V). Such a material is said to be ohmic, and we write Ohm’s Law:

I

VR

IRV

Resistivity

An object which provides resistance to current flow is called a resistor. The actual resistance depends on:

• properties of the material

• the geometry (size and shape)

The symbol for a resistor is

For a conductor of length L and area A, the resistance is

where is called the resistivity of the material.A

LR

Temperature Dependence and Superconductivity

In general, the resistivity of most materials will depend on the temperature. For most metals, resistivity increases linearly with temperature:

Some materials, when very cold, have a resistivity which abruptly drops to zero. Such materials are called superconductors.

)](1[ 00 TT

A bird lands on a bare copper wire carrying a current of 32 A. The wire is 8 gauge, which means that its cross-sectional area is 0.13 cm2. (a) Find the difference in potential between the bird’s feet, assuming they are separated by a distance of 6.0 cm. (b) Will your answer to part (a) increase or decrease if the separation between the bird’s feet increases?

Direct Current (DC) Circuits

A circuit is a loop comprised of elements like resistors and capacitors around which current flows.

For current to continue to flow in a circuit, there must be an energy source such as a battery.

The light bulb in this circuit is the resistor. Connecting wires are assumed to have zero resistance.

battery

battery

Electric potential increases by

Electric potential decreases by IR

I

- +

R

I

Battery as emf in DC Circuits

+ terminal at higher potential than – terminal

Imagine positive charges moving clockwise around the circuit. The electric potential increases by 12 V across the battery and decreases by 12 V across the resistor.

Energy and Power in Electric Circuits

Resistance is like an internal friction; energy is dissipated. The energy dissipated per unit time is the power P:

P =U/ t =(Q/t)V = IV

SI unit: watt, W

Using Ohm’s Law, V=IR, power can be rewritten as:

P = I2R = V2/R

Energy Usage:

1 kilowatt-hour = (1000 W)(3600 s) = (1000 J/s)(3600 s) = 3.6106 J

It costs 2.6 cents to charge a car battery at a voltage of 12 V and a current of 15 A for 120 minutes. What is the cost of electrical energy per kilowatt-hour at this location?

A 75-W light bulb operates on a potential difference of 95 V. Find the current in the bulb and its resistance.

Resistors in Series and Parallel

Any two circuit elements can be combined in two different ways:

• in series - with one right after the other; the same current must flow through both elements.

• in parallel – connected across the same potential difference; the current is divided into two paths.

Series Combination

R1 R2

Parallel Combination

R1

R2

The current I is the same in both The current may be different in resistors, so the voltage Vba must each resistor, but the voltage satisfy:

Vba is the same across each resistor and the total current

is conserved: I = I1 + I2

Series Combination

R1 R2

a b

Parallel Combination

R1

R2a b

Equivalent Resistance

21

111

RRReq

Vba= IR1 + IR2 = I(R1 + R2)

Req = R1+ R2

(a) Three resistors, R1, R2, and R3, connected in series. Note that the same current I flows through each resistor.

(b) The equivalent resistance, Req = R1 + R2 + R3

has the same current flowing through it as the current I in the original circuit.

Resistors in series

(a) Three resistors, R1, R2, and R3, connected in parallel. Note that each resistor is connected across the same potential difference, .

(b) The equivalent resistance,

has the same current flowing through it as the total current I in the original circuit.

Resistors in parallel

321

1111

RRRReq

Consider the circuit shown in the figure, in which three lights, each with a resistance R, are connected in parallel. What happens to the intensity of light 3 when the switch is closed? What happens to the intensities of lights 1 and 2?

Conceptual Question

(a) The two vertical resistors are in parallel with one another, hence they can be replaced with their equivalent resistance, R/2.

(b) Now, the circuit consists of three resistors in series. The equivalent resistance of these three resistors is 2.5 R.

(c) The original circuit reduced to a single equivalent resistance.

Analyzing a complex circuit of resistorsAll resistors are the same in Figure (a).

What is the equivalent resistance?

Walker Problem 26, pg. 710

The current in the 13.8 resistor is 0.750 A. Find the current in the other resistors in the circuit.

Walker Problem 44, pg. 711

Kirchhoff’s Rules

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 junctionequals 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.

0I

0V

follows from conservation of charge

follows from conservation of energy

Kirchhoff’s junction rule states that the sum of the currents entering a junction must equal the sum of the currents leaving the junction. In this case, for the junction labeled A:

I1 = I2 + I3 or I1 – I2 – I3 = 0

Kirchhoff’s junction rule

Applying Kirchhoff’s junction rule to the junction A:

I1 I2 I3 = 0

I3 = (2.0 5.5) A = 3.5 A

The minus sign indicates that I3 flows opposite to the direction shown; that is, I3 is upward.

A specific application of Kirchhoff’s junction rule

Kirchhoff’s loop rule states that as one moves around a closed loop in a circuit the algebraic sum of all potential differences must be zero. The electric potential:

• increases as one moves from the minus to the plus plate of a battery • decreases as one moves through a resistor in the direction of the current

Kirchhoff’s loop rule

Junction Rule: I1 = I2 + I3

Analyzing a simple circuit

I3R I1R=0 I3R I2R = 0 What is the equation?

Loop Rule: Use any two of these three loops

How much current flows through each battery when the switch is (a) closed and (b) open? (c) With the switch open, suppose that point A is grounded. What is the potential at point B?

Walker Problem 52, pg. 711

A

B

Capacitors are used in electronic circuits. The symbol for a capacitor is

We can also combine separate capacitors into one effective or equivalent capacitor. For example, two capacitors can be combined either in parallel or in series. Series

ParallelCombination Combination

C1 C2

C1

C2

Circuits containing Capacitors

+

(a) Three capacitors, C1, C2, and C3, connected in parallel. Note that each capacitor is connected across the same potential difference, .

(b) The equivalent capacitance,

Ceq = C1 + C2 + C3

has the same charge on its plates as the total charge on the three original capacitors.

Capacitors in parallel

(a) Three capacitors, C1, C2, and C3, connected in series. Note that each capacitor has the same magnitude of charge on its plates.

(b) The equivalent capacitance,

has the same charge as the original capacitors.

Capacitors in series

321

1111

CCCCeq

Parallel vs. Series Combination

Parallel Series

• charge Q1 , Q2 • charge on each is Q

• total Q = Q1 + Q2 • total charge is Q

• voltage on each is V • voltage V1 , V2

• Q1= C1V • Q = C1V1

• Q2= C2V • Q = C2V2

• Q = CeffV • Q = Ceff(V1+V2)

• Ceff = C1+C2 • 1/Ceff = 1/C1+1/C2

Walker Problem 54, pg. 711

A 15 V battery is connected to three capacitors in series. The capacitors have the following capacitance: 4.5 F, 12 F, and 32 F. Find the voltage across the 32 F capacitor.

RC Circuits

We can construct circuits with more than just a resistor. For example, we can have a resistor, a capacitor, and a switch:

The capacitor acts like an open circuit: no charge flows across the gap. However, when the switch is closed, current can flow from the negative plate of the capacitor to the positive plate.

R

C

S

When the switch is closed the current will change.

(a) Before the switch is closed (t < 0) there is no current in the

circuit and no charge on the capacitor.

(b) After the switch is closed (t > 0) current flows and the charge on the capacitor builds up over a finite time. As t increases without limit, the charge on the capacitor approaches Q = C.

A typical RC circuit

Capacitor Charging

Assume that at time t = 0, the capacitor is uncharged, and we close the switch. It can be shown that the charge on the capacitor at some later time t is:

q = qmax(1 – e-t/)

The time constant =RC, and qmax is the maximum amount of charge that the capacitor will acquire: qmax=C

The current is given by

I = (/R)e-t/

Charge versus time for an RC circuit

Current versus time for an RC circuit

What happens after the switch is closed?

The capacitor is initially uncharged.

Walker Problem 62, pg. 712The capacitor in an RC circuit (R = 120 , C = 45 F) is initially uncharged. Find (a) the charge on the capacitor and (b) the current in the circuit one time constant ( = RC) after the circuit is connected to a 9.0 V battery.

Consider the circuit shown below. (a) Is the current flowing through the battery immediately after the switch is closed greater than, less than, or the same as the current flowing through the battery long after the switch is closed? (b) Find the current flowing through the battery immediately after the switch is closed. (c) Find the current in the battery long after the switch is closed.

Walker Problem 78, pg. 713

(a) A charged capacitor is connected to a resistor. Initially the circuit is open, and no current can flow.

(b) When the switch is closed current flows from the + plate of the capacitor to the - plate. The charge remaining on the capacitor approaches zero after several time units, RC.

Discharging a capacitor

Capacitor Discharging

Consider this circuit with a charged capacitor at time t = 0:

It can be shown that the charge on the capacitor is given by:

q(t) = Qe-t/

The time constant = RC.

Current versus time in an RC circuit

R

C

S

+Q

-Q

To measure the current flowing between points A and B in (a) an ammeter is inserted into the circuit, as shown in (b). An ideal ammeter would have zero resistance.

Measuring the current in a circuit

An ammeter is device for measuring currents in electrical circuits.

Measuring the voltage in a circuit

The voltage difference between points C and D can be measured by connecting a voltmeter in parallel to the original circuit. An ideal voltmeter would have infinite resistance.

A voltmeter measures voltage differences in electrical circuits.