PHYSICS 3 (ELECTRICITY AND MAGNETISM) - CHAPTER 3 potx

Chapter 3 CURRENT AND RESISTANCE, DIRECT CURRENT CIRCUITS 3.1 Electric Current 1 The electric current in a conductor is defined by dq i = here dq is the amount of positive charge that pa

Chapter 3 CURRENT AND RESISTANCE, DIRECT CURRENT CIRCUITS

3.1 Electric Current

1) The electric current in a conductor is defined by

dq

i =

here dq is the amount of (positive) charge that passes in time dt through a hypothetical surface that cuts across the conductor By convention, the direction of electric current is taken as the direction in which positive charge carriers would move The SI unit of electric current is ampere (A) : 1A = 1C/s

2) The current i (a scalar) is related to the current density J

r

(a vector) by

where dA

uuur

is a vector perpendicular to a surface element of area dA and the integral is taken over any surface cutting across the conductor.J

r

has the same direction as the velocity of the moving charges if they are positive and the opposite direction if they are negative

3.2 A Model for Electrical Conduction

When a conductor does not have a current through it, its conduction electrons move randomly, with no net motion in any direction When the conductor has a current through it, these electrons still move randomly, but now they tend to drift with a drift speed vd in the direction opposite that of the applied electric field that causes the current The drift speed is tiny compared with the speeds in the random motion For example, in the copper conductors of house-hold wiring, electron drift speed are perhaps 10-5 or 10-4 m/s, where as the random-motion speeds are around 106 m/s

(a) (b) (c)

Fig 3.1 : Random motion of an electron from A to F (the electron collides with an atom at B, C, D, E)

a: without electric field

b: in presence of an electric field E, the electron drifts rightward

c: superposition of figure a and figure b

Consider a wire of length L, cross-sectional area A, number of carriers (free electrons) per unit volume n The total charge of the wire is

(e = 1.602 x 10-19 C) Since the free electrons drift along the wire with speed vd (in the direction opposite that of the current i), the total charge q moves through any cross section of the wire in the time interval

t =

d

L

and the current i, which is the time rate of transfer of charge across a cross section, is given by

i = q

The current density J

r

(current per unit sectional area) is given by

J

r

Note that the minus sign in (3.5) and (3.6) implies that the direction of the current i is opposite to that of the drift of the free electrons in the wire

Example : Consider a copper wire which carries a current i = 17mA Let r = 900µm be the radius of the wire Assume that each copper atom contributes one conduction electron to the current and that the current density is uniform across the wire cross section The drift speed of the conduction electrons can be determined from (3.5)

vd = - i

neA =

J

Since each copper atom contributes one conduction electron to the current, the number n of conduction electrons per unit volume is the same as the number of atoms per unit volume

n = number of atoms per unit volume

= (number of atoms per mole)x(number of moles per unit mass)x(mass per unit volume)

number of atoms per mole = Avogadro’s number = NA = 6.02x1023

number of moles per unit mass = inverse of the mass per mole of copper M

= 63.54 g/mol = 63.54x10-3 kg/mol mass per unit volume = mass density of copper ρmass = 8.96 g/cm3 = 8.96x103 kg/m3

n = NAρmass/M = 6.02x1023x8.96x103/63.54x10-3 = 0.8489x1029 electrons/m3

The current density : J = 17x10-3/(πr2) A/m2

The charge of an electron : e = 1.602 x 10-19 C

⇒ vd = - J

ne = -4.9x10

-7 m/s

3.3 Resistance and Ohm’s Law

Ohm’s law

i

V

where V is the potential difference across the conductor and i is the current

Fig 3.2 Resistivity ρ and conductivity σ of a material

J

E

1 = σ

=

Vector form

J E

r r

ρ

The resistance of a conducting wire of length L and uniform cross section is

A

L

R =ρ

where A is the cross-sectional area

Change of ρ with temperature : for many materials, including metals, the relation between ρ and temperature T is approximated by

where ρo is the resistivity at temperature To, α is the temperature coefficient of resistivity for the material Resistivity of a metal

τ

= ρ

n e

m

here n is the number of free electrons per unit volume and τ is the mean time between collisions of an electron with the atoms of the metal

3.4 Electrical Energy and Power

Rate of electrical energy transfer

Resistive dissipation

In a resistor, electric potential energy is converted to internal thermal energy via collisions between charge carriers and atoms

3.5 Electromotive Force (EMF)

The electromotive force of a device is the work the device does to force a unit positive charge from the negative to the positive terminal

dq

dW

3.6 Kirchoff’s Rules

Loop rule: The algebraic sum of the changes in potential encountered in a complete traversal of any loop

of a circuit must be zero

Junction rule

The sum of the currents entering any junction must be equal to the sum of the currents leaving that junction

Single loop circuits (Fig.3.3) :

R

r +

ε

=

According to the Loop Rule the potential difference caused by the battery ( ) must be compensated for by the potential drops across the two resistors (r and R) in Fig 3.3 Notice that the potential (V) starts at Va and then returns again to Va after resistor R (Fig 3.4)

Power

3.7 Resistors in Series and in Parallel

Series resistances

Parallel resistances

=

i

1 R

1

(3.20)

3.8 RC Circuits (Fig 3.5)

1) Charging a capacitor

e = Ri + V = R

dt

dq

+

C

1

q ⇒

dt

dq

+

RC

1

q =

R

1

e

Let x = q - Ce ⇒

dt

dx

+

RC

1

x = 0 ⇒

x

dx

=

-RC

dt

⇒ ln(x) =

-RC

t

+ const ⇒ x = A RC

t

e

−

t

e

−

since q(0) = 0 ⇒ A = -Ce

t

e

−

i =

R

e RCt e

−

Fig 3.5 2) Discharging a capacitor

0 = Ri + V = R

dt

dq

+

C

1

q ⇒

dt

dq

+

RC

1

q = 0 ⇒

q

dq

=

-RC

dt

⇒ q = A RC

t

e

−

since q(0) = Ce ⇒ A = Ce

t

e

−

i =

-R

e RCt e

−

The negative sign indicates that the current flows in the opposite direction

The quantity τ = RC is called the time constant It dictates the rate of voltage build up on the capacitor, and the rate of current decrease

Problems

Electric current

3.1) An isolated conducting sphere has a 10cm radius One wire carries a current of 1.000.002 A into it Another wire carries a current of 1.000.000 A out of it How long would it take for the sphere to increase

in potential by 1000 V ?

3.2) A lightning of current I = 100kA strikes the ground at point O (Fig P3.1) The current spreads through the ground uniformly over a hemisphere centered on the strike point The resitivity of the ground is ρ = 100Ωm Find the potential difference between A and B The radial distance OA = 60m, OB = 62m

Solution : J =

2

r 2

I

π ⇒ E = ρJ = 2 r2

I π

ρ

⇒ VAB = -OB∫

OA

Edr

3.3) Consider the circuit in Fig P3.2 with e(t) = 12sin(120πt) V, r = 10Ω Find the value of R such that the power in R is maximized ?

Circuit

3.4) A 9.0 volt battery is connected across a light bulb (R = 3.0 ) How many electrons pass through the resistor in one minute? How many joules of energy are generated in one minute?"

3.5) A battery has an initial internal resistance of 0.75 and an emf of 9 V It is placed a cross a 5Ω resistor and a 10 µF capacitor hooked up in parallel

a) After a the capacitor has charged, what is the current through the resistor?

b) What is the charge on the capacitor?

c) If the battery is disconnected, how long will it take the capacitor to reach one-third of its initial voltage?"

3.6) The capacitor C in Fig P3.3 is initially uncharged At t = 0, the switch K is closed Determine an expression for the potential difference V and the current i of the circuit

3.7) In Fig P3.4, e1 = 12V, e2 = 24V, r1 = 10Ω, r2 = 5Ω, R = 2Ω Determine i1, i2, i

3.8) The circuit in Fig P3.5 has E = 12V, R1 = 10Ω, R2 = 30Ω, r = 5Ω Find the currents i1, i2, i

3.9) The circuit in Fig P3.6 has e1 = 12V, e2 = 6V, e3 = 9V, r1 = 4Ω, r2 = 3Ω, r3 = 2Ω Find the currents i1, i2,

i

3.10) The capacitor C in Fig P3.7 is initially uncharged At t = 0, the switch K is closed Determine an expression for the potential difference V and the current i of the circuit

Homeworks 3

H3.1 The capacitor C in Fig H3.1 is initially uncharged At t = 0, the switch K is closed Determine an expression for the potential difference V and the current i of the circuit (e in [V], r and R in [Ω], C in [µF])

R 300 450 600 750 900 300 450 600 750 900 300 450 600 750 900 300

R 300 450 600 750 900 300 450 600 750 900 300 450 600 750 900 300

R 300 450 600 750 900 300 450 600 750 900 300 450 600 750 900 300

R 300 450 600 750 900 300 450 600 750 900 300 450 600 750 900 300

H3.2 Determine the currents i, i1, i2 in Fig H3.2 (e1 and e2 in [V], r1, r2 and R in [Ω])