1. Current and Resistance
Chapter 27:
Current and Resistance
27.1 Electric Current
27.2 Resistance and Ohm’s Law
27.4 Resistance and Temperature
27.6 Electrical Energy and Power
Fig 27-CO These power lines transfer energy from the power company to homes and businesses. The energy is transferred at a very high voltage, possibly hundreds of thousands of volts in some cases. Despite the fact that this makes power lines very dangerous, the high voltage results in less loss of power due to resistance in the wires. (Telegraph Colour Library/FPG)
Fig 27-CO, p.831

2. 27-1 Electric Current
Consider a system of electric charges in motion. Whenever there is a net flow of charge through a region, a current is said to exist.
The charges are moving perpendicular to a surface of area A,
The current is the rate at which charge flows through this surface.
average current (Iav)
Instantaneous current (I)

3. + - The SI unit of current is the ampere (A):
That is, 1 A of current is equivalent to 1 C of charge passing through the surface area in 1 s.
It is conventional to assign to the current is the same direction as the flow of positive charge +
Current -
The direction of the current is opposite the direction of flow of electrons.
It is common to refer to a moving charge (positive or negative) as a mobile charge carrier. For example, the mobile charge carriers in a metal are electrons.

4. Microscopic Model of Current
Let ΔQ = number of charge carriers in section x
ΔQ = (n q) A Δ x
ΔQ = n q A (vd Δ t)
Because Δx = (vd Δ t)
Figure 27.2 A section of a uniform conductor of cross-sectional area A. The mobile charge carriers move with a speed vd , and the displacement they experience in the x direction in a time interval Δt is x vd Δt. If we choose t to be the time interval during which the charges are displaced, on the average, by the length of the cylinder, the number of carriers in the section of length Δx is nAvd Δt, where n is the number of carriers per unit volume.
Ampere
The speed of the charge carriers vd is an average speed called the drift speed.
Fig 27-2, p.833

5. Consider a conductor in which the charge carriers are free electrons.
If the conductor is isolated (not connected with battery), the potential difference across the conductor is zero , these electrons moved with random motion that is similar to the motion of gas molecules.
When a potential difference is applied across the conductor (for example, battery), an electric field is set up in the conductor; this field exerts an electric force on the electrons, producing a current.

6. However, the electrons do not move in straight lines along the conductor. Instead, they collide repeatedly with the metal atoms, and their resultant motion is complicated and zigzag (Fig. 27.3). Despite the collisions, the electrons move slowly along the conductor (in a direction opposite that of E) at the drift velocity vd . +
Figure 27.3 A schematic representation of the zigzag motion of an electron in a conductor. The changes in direction are the result of collisions between the electron and atoms in the conductor. Note that the net motion of the electron is opposite the direction of the electric field. Because of the acceleration of the charge carriers due to the electric force, the paths are actually parabolic. However, the drift speed is much smaller than the average speed, so the parabolic shape is not visible on this scale.
Fig 27-3, p.834

7. Ex1
The electric current when an electric charge of 5 C passes an area each 10-3 sec is:
= 5/10-3 = 5000 A
Ex2
In a particular cathode ray tube, the measured beam current is 30 A . The number of electrons strikes the tube screen every 40 sec
Ex3
If the current carried by a conductor is doubled, the electron drift velocity is

8. where  is conductivity
Consider a conductor of cross-sectional area A carrying a current I. The current density J in the conductor is defined as the current per unit area where
where current I = n q v d A
A/m2
A current density J and an electric field E are established in a conductor when a potential difference is maintained across the conductor.
If the potential difference is constant, then the current also is constant. In some materials, the current density is proportional to the electric field:
where  is conductivity

9. Ohm’s law states that ; Ohm’s law
where the constant of proportionality σ is called the conductivity of the conductor. Materials that obey Equation are said to follow Ohm’s law
Ohm’s law states that ;
For many materials (including most metals), the ratio of the current density J to the electric field E is equal a constant σ
Ohm’s law
Figure 27.5 A uniform conductor of length and cross-sectional area A. A potential
difference ΔV =Vb-Va maintained across the conductor sets up an electric field E,
and this field produces a current I that is proportional to the potential difference.
mass m. This can be written as
Materials that obey Ohm’s law are said to be ohmic,
Materials that do not obey Ohm’s law are said to be non-ohmic
Fig 27-5, p.836

10. Important relation Ohm`s law
We can obtain a general form of Ohm’s law by considering a segment of straight wire of uniform cross-sectional area A and length 
If the field is assumed to be uniform, the potential difference is related to the field through the relationship
Important relation
Ohm`s law
R is the resistance of the conductor

11. Other Important relation
The inverse of conductivity is resistivity ()
where the SI unit of  is (ohm . m) or (.m)
Other Important relation
Ohm`s law
From this result we see that resistance has SI units of volts per ampere. One volt per ampere is defined to be 1 ohm (Ω): Ex
The ratio of an electric potential across a resistor to the passing current is

12. (a) The current–potential difference curve for an ohmic material
(a) The current–potential difference curve for an ohmic material. The curve is linear, and the slope is equal to the inverse of the resistance of the conductor.
(b) A nonlinear current–potential difference curve for a semiconducting diode. This device does not obey Ohm’s law.
Figure 27.7 (a) The current–potential difference curve for an ohmic material. The curve is linear, and the slope is equal to the inverse of the resistance of the conductor.
Fig 27-7a, p.838

13. Table 27-1, p.837

14. Geometric shapes of resistors
An assortment of resistors used in electrical circuits.
p.837

15. Figure 27.6 The colored bands on a resistor represent a code for determining resistance. The first two colors give the first two digits in the resistance value. The third color represents the power of ten for the multiplier of the resistance value. The last color is the tolerance of the resistance value. As an example, the four colors on the circled resistors are red (2), black (0), orange (10 3 ), and gold (5%), and so the resistance value is 20x103Ω =20 k Ω with a tolerance value of 5% 1 k. (The values for the colors are from Table 27.2.)
Fig 27-6, p.838

16. Table 27-2, p.838

17. a) Calculate the resistance R of an aluminum cylinder that is
10 cm long and has a cross-sectional area of 2 x 10-4 m b) Repeat the calculation for a cylinder of the same dimensions and made of glass having a resistivity of 3x1010 Ω

18. Ex1
If 1.0 V potential difference is maintained across a 1.5 m length of tungsten wire ( = 5.7 x10-8 ohm.m) that has a cross-sectional area of 0.6 mm2. the current in the wire (7A) Ex
An aluminum wire having a cross sectional area of 4 x10-6 m2 carries a current of A, the current density is
And if the drift speed of the electron in the wire is 0.13 mm/s. the number of charge carriers per unit volume is

19. Ex
A resistor is constructed of a carbon rod (  = 3.5 x10-5  m) that has a uniform cross-sectional area of 5 mm2. when a potential difference of 15 V is applied across the ends of the rod, the carriers a current of 4 A. the rod`s length is

20. 27-3 RESISTANCE AND TEMPERATURE
Over a limited temperature range, the resistivity of a metal varies approximately linearly with temperature according to the expression
where ρ is the resistivity at some temperature T (in degrees Celsius), ρ0 is the resistivity at some reference temperature T0 (usually taken to be 20°C), and α is the temperature coefficient of resistivity.

21. Temperature coefficient of resistivity ()
The SI unit of () is
Because resistance is proportional to resistivity (Eq ), we can write the variation of resistance as
R= Ro + Ro  (T-To)  R = R- Ro = Ro  (T-To)

22. A resistance thermometer, which measures temperature by measuring the change in resistance of a conductor, is made from platinum and has a resistance of 50 Ω at 20°C. When immersed in a vessel containing melting indium, its resistance increases to 76.8 Ω . Calculate the melting point of the indium.

23. R= Ro + Ro  (T-To)  R = R- Ro = Ro  (T-To)Ex
The fractional change in resistance ( R/Ro) of an iron wire ( =5 x10-3 oC-1) when the temperature changes from 20 oC to 60 oC is
R= Ro + Ro  (T-To)  R = R- Ro = Ro  (T-To)

24. If a battery is used to establish an electric current in a conductor, the chemical energy stored in the battery is continuously transformed into kinetic energy of the charge carriers.
In the conductor, this kinetic energy is quickly lost as a result of collisions between the charge carriers and the atoms making up the conductor, and this leads to an increase in the temperature of the conductor.
In other words, the chemical energy stored in the battery is continuously transformed to internal energy associated with the temperature of the conductor.

25. Now imagine the following :
A positive quantity of charge q that is moving clockwise around the circuit from point b through the battery and resistor back to point a.
As the charge q moves from a to b through the battery, its electric potential energy U increases by an amount U= ΔV Δ q (where Δ V is the potential difference between b and a
However, as the charge moves from c to d through the resistor R, it loses this electric potential energy U due to collide with atoms in the resistor R, producing an internal energy. If we neglect the resistance of the connecting wires, no loss in energy occurs for paths bc and da. When the charge arrives at point a,
Active Figure A circuit consisting of a resistor of resistance R and a battery having a potential difference V across its terminals. Positive charge flows in the clockwise direction.
Fig 27-13, p.845

26. The rate of loses of potential energy through the resistor is
Because the rate at which the charge loses energy equals the power P delivered to the resistor (which appears as internal energy), we have
When I is expressed in amperes, V in volts, and R in ohms,
the SI unit of power is the watt

27. An electric heater is constructed by applying a potential difference of 120 V to a Ni-chrome wire that has a total resistance of 8.0 Ω . Find the current carried by the wire and the power rating of the heater.
If we doubled the applied potential difference, the current would double but the power would quadruple because

28. Estimate the cost of cooking a turkey for 4 h in an oven that operates continuously at 20.0 A and 240 V.

29. EX
If the cost of electricity in SA is SR/kW-h , then the cost of leaving a 60 W light lamp ON for 14 days is
Total Power= 60 x14x24 = W= kW
The cost = 2.016x0.05= SR Ex
An electric heater is operated with a potential difference of 110 V to a Tungsten
Wire that has a resistance of 10  . The power of the heaters: