Unit 7 Electricity I

Unit 7 Electricity I

1. Electrostatics 1.1 Electric charge If a glass rod is rubbed with a silk, some electrons are moved from the glass to the silk by friction. So the glass rod becomes positively charged and the silk negatively charged. Since the glass and silk have opposite charges, they attract each others. Two such rubbed glass rods will repel one another since each rod has positive charges on it. The phenomenon of attraction or repulsion between charged objects is called electrostatics. In above process, no electrons are created or destroyed; they are simply transferred from one object to another, while the total charges are not changed. The rules of electrostatics are: • objects with the same charge repel • objects with opposite charges attract

Electric charge q has the unit Coulomb in SI unit, abbreviated C. The basic charge of an electron and a proton are identical in magnitude but opposite in sign. The magnitude of this basic charge is: qe =1.6x10-19 C. From above relation: 1 C = 6.25 x 1018 electrons (or protons)

1.2 Electrostatic force The attraction or repulsion forces between charges are called electrostatic force. The direction of the electrostatic force is based on the charge involved. Opposite charges generate an attractive force (the sign is negative); like charges generate a repulsive force (the sign is positive). The magnitude of the force is proportional to the product of q1 and q2 ; and inversely proportional to the square of the distance r. Therefore the electrostatic force: • decreases with distance  increases with quantity of charges

r

q1

r

q2 (a) Opposite charges

q1

q2 (b) like charges 1

1.2 Charges at work Electrons can be transferred from one object to another by simple touching; this method of charging is called charging by contact. The example of electroscope is an example of charging by contact. An electroscope is made with gold foil leaves hung from a metal stem and is insulated from the air in a glass-walled container. (a) A positively charged glass rod is brought near the electroscope, attracting electrons to the top and leaving a net positive charge on the leaves. Like charges in the light flexible gold leaves repel, separating them. (b) When the rod is touched against the ball, electrons are attracted and transferred, reducing the net charge on the glass rod but leaving the electroscope positively charged. (c) The excess charges are evenly distributed in the stem and leaves of the electroscope once the glass rod is removed. Substances, such as metals, in which electrons can flow freely, are good conductors. Plastic, polyethylene and rubber conduct badly and are called insulators. Charging by contact is not the only way to transfer excess charges to a metallic object in order to charge it. In this example, (a) Two uncharged or neutral metal spheres are in contact with each other but insulated from the rest of the world. (b) A positively charged glass rod is brought near the sphere on the left, attracting negative charge and leaving the other sphere positively charged. (c) The spheres are separated before the rod is removed, thus separating negative and positively charge. (d) The spheres retain net charges after the inducing rod is removed. This process is called charging by induction.

In insulators the charging takes place differently: no free charge flows, a charged object attract an insulator by polarizing its molecules: the center of positive charge and negative charge of a molecules shift slightly: charge polarization.

Negatively charged clouds induce a positive charge on the surface of a building. Lightening strike occurs when there is a sudden discharge between the cloud and the building. The purpose of a 2

lightening rod is to continually collect electrons from air and discharge them, preventing a large buildup of positive charges on the building by induction.

1.3 The electric field

Electric field E defines the electric force exerted on a positive test charge qo positioned within any given space:

r

F E= qo

q

qo

The direction of E depends on the force exerted by the charge q. Since the test charge qo is positive, if the charge q is positive, an repulsive force is generated and the direction of E is away the charge q; if the charge q is negative, an attractive force is generated and the direction of E is toward the q;. Electric field is expressed in SI unit as N/C. Electric field line is used to express directly the direction and the intensity of the electric field E. The arrow points to the direction of the electric field, the number of lines indicates the intensity of the electric field.

+

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1.4 The electric potential difference (voltage) A charge accelerated by an electric field is analogous to a mass going down hill. In both cases potential energy is converted to kinetic energy. In the first case; the electric potential energy (EPE) of the charge is converted to kinetic energy of the charge q.

The electric potential difference or voltage, ∆V , is the change of the EPE per unit charge, considering the test charge q moving from A to B, then: ∆V = VB −VA =

EPE B − EPE q

A

or

∆EPE = ∆V ⋅ q

The unit for ∆V is Volt (V). 1V=1 J/C.

1.5 Capacitors A capacitor is a device that can store energy and consists of two oppositely charged conductor plates that are separated by a distance d. The capacitance C is defined as :

C =

q ∆V

where q is the magnitude of charge stored on each plate and ∆V is the potential difference between the two metal plates. (sometimes we use simply V to represent

∆V

+Q

).

-Q E

Area A

d

The SI unit for capacitance is the farad (F), 1 F =1 C/V. The potential energy stored in a capacitor is given by : U =

1 c( ∆V ) 2 2

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The insulating material inserted between the plates is called a dielectric. The charged plates polarize the molecules in the insulating material between the plates of a capacitor. This produces a layer of opposite charge on the surface of the dielectric that attracts more charge onto the plate, increasing its capacitance. The capacitance of a parallel plate capacitor with a dielectric is:

C =

κ εo A d

−12 2 2 where ε o = 8.85 × 10 C / N ⋅ m is the permittivity of the free space,

of the insulating material, and A is the area of each plate.

κ the dielectric constant

All living cells are protected from environment by a thin semi-permeable wall called a cell membrane. The membrane possesses channels of pores that allow a selective passage of metabolites and ions in and out of the cell. The thickness of a cell membrane varies between 7nm to 10nm (1nm=10-9m). Membranes act as capacitor maintaining a potential difference between oppositely charged surfaces of inside and out cell. This potential is about 0.1V, giving rise to an electric field of about 10MV/m, a very high electric field. A typical value of the capacitance per unit area C/A is about 1 mF/cm2 for cell membranes. This relates the membrane’s dielectric constant via:

κ

giving a value of Materials: :

κ

C κ oε = A d

κ ≅ 10 for membrane. Here is some dielectric constants of various dielectrics:

Air 1

Pyrex mica 4.7 5.4

Silicon 12

Ethanol 25

water 80

BaTiO3 7000

2. Electric current 2.1 Charge flow A charged particle is accelerated in a electric field. The flow of electric charges in a conductor is called electric current. The flow of charges is caused uniquely by the electric potential difference (or voltage) at the two ends of the conducting wire. A positive charge accelerates from a region of higher electric potential to a region of lower potential; a negative charge accelerates from a region of lower potential to a region of higher potential. To sustain such flow of charge, a special device must be provided to maintain the voltage. The situation is analogue to the flow of water from a higher reservoir to a lower one. The simplest “pumping” device which provides such voltage is a battery.

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The current I is given by:

I =

∆q ∆t

where ∆ q is the magnetite of the charge crossing a surface in a time ∆ t, the surface being perpendicular to the motion of charge. The SI unit of current is ampere (A), 1 A= 1 C/s. In a circuit of metal wires, electrons make-up the flow of charge. These electrons are called conduction electrons. The direction of the current is defined as the opposite direction of the conduction electrons.

2.2 Electrolysis Certain chemical compounds conduct electricity when they are melted or dissolved in water. Solution containing salt and water is an electrolyte. In solution, sodium chloride will ionize giving + sodium ion Na and chloride ion Cl-: NaCl → Na + + Cl − and Cl − ions are mobile ions and can move if a voltage is applied to the solution by means of two electrodes: an electric current is formed by negatively charged chloride ions Cl − + moving to the positive electrode (anode), and positively charged sodium ions Na to the negative electrode (cathode). The conducting liquid is called electrolyte and the process is called electrolysis. Ohm’s law reminds valid for electrolytic current. Na

+

Electrical current in organisms is generally not carried by electrons. Instead it is carried by the mobile ions, such as Na+, Cl-, K+ of electrolytic solutions. Ohm’s law reminds valid for electrolytic current. The typical order of resistivity for body fluid is about 1 Ω ⋅ m . This is eight order of magnitudes of the resistivity of copper.

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When a small dc current is applied to the body by means of two electrodes, one cathode and another anode, the body tissue will carry on the current. This happens because tissue fluid is an electrolyte and it contains a high percentage of salt ions, such as Na+, Cl-, K+. This is the basic principle of galvanic treatment used in beauty therapy.

2.3 Electrical resistance The rate of charge flow, the current, depends on the voltage and the electrical resistance provided by the conductor. Their relationship is summarized by the Ohm’s law , V I = R It states that the current in a circuit is directly proportional to the voltage established across the circuit and is inversely proportional to the resistance of the circuit: The resistance is measured in volts per ampere, a unit called ohm ( Ω). Typical light bulb has a resistance of 100 ohms, an iron or toaster has a resistance of 15-20 ohms. Inside a circuit, current is regulated by a device called resistors. Rules for resistors: • Resistance varies with temperature. For metals it increases with temperature. • Resistance depends on the nature of substance • For a resistor of given shape, its resistance is directly proportional to length (which is also the direction of the current path) and is inversely proportional to the cross section area A: L R =ρ A R is the resistance in ohms, L length in m, A area in m2. ρ resistivity in ohm-m.

A

A high resistivity indicates that the material is a good insulator, while a low resistivity means that the material is a good conductor.

L

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------------------------------------------------------------------------Material Resistivity (Ω ．m) Characteristic -----------------------------------------------------------------------Copper 1.7x10-8 Good conductor Germanium 0.6 Fair conductor Body fluids 2-0.2 Fair conductor 9~14 Glass 10 Good insulator Mica 1011-15 Good insulator ------------------------------------------------------------------------

2.4 Electric circuit The equivalent resistance R of a series combination of resistance R1, R2, R3 …is : R= R1+ R2+ R3 +… The equivalent resistance R of a parallel combination of resistance R1, R2, R3 …is :

1 1 1 1 = + + + ... R R1 R2 R3

2.5 Electric power A current moving in circuit converts its potential energy to other form of energies: thermal energy, mechanical energy, light energy…. It is proportional to electric potential difference ∆ V and charge q: i.e. ∆EPE = ∆V ⋅ q

The rate at which electric energy is converted into other form of energy is called electric power. i.e. ∆EPE ∆V ⋅ q P= = = ∆V ⋅ I t t ∆ V can be simplified to voltage V. Thus electric power P is equal to the product of current and voltage: P = IV Power has the unit of J/s, or watts (W), thus 1 A x 1V =1W. When current passes through a resistor R, the electric energy is converted to other energy and is related to the resistance by Ohm’s law: V2 P = I 2R = R

2.6 Direct current (dc) and alternating current (ac)

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When the charge flow is uni-directional, the current is called direct current (dc), If the direction of the charge flow changes from moment to moment, the current is referred to as alternating current (ac). In a circuit, if the current is dc, both voltage and current are constant in time; if the current is ac, the voltage and current vary sinusoidally with time. V =V0 sin 2πft I = I 0 sin 2πft

The heights of the waves give peak current Io and peak voltage Vo. and Io=Vo/R. In HK, the frequency of the ac mains is 60 Hz, that is, the current varies 60 cycles in 1 s.

2.7 Average power and peak power 2 The power of an ac current is P = IV = I oV sin 2πf and is

time-varying. Its peak value is IoVo. So the average power ∴Pave =

1 peak power = 12 I 0V0 2

Similarly we define average current (rms or effective) and average voltage of an ac current:

I rms = From above:

I0 2

Pave = I rms Vrms =

Vrms =

V0 2

I o Vo 1 = I oVo 2 2 2

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