 Electric Currents and Resistance

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Electric Currents and Resistance
Chapter 25 opener. The glow of the thin wire filament of a light bulb is caused by the electric current passing through it. Electric energy is transformed to thermal energy (via collisions between moving electrons and atoms of the wire), which causes the wire’s temperature to become so high that it glows. Electric current and electric power in electric circuits are of basic importance in everyday life. We examine both dc and ac in this Chapter, and include the microscopic analysis of electric current.

The Electric Battery Electric Current Ohm’s Law: Resistance and Resistors Resistivity Electric Power Microscopic View of Electric Current: Current Density and Drift Velocity

Power in Household Circuits
Microscopic View of Electric Current: Current Density and Drift Velocity Superconductivity*

This is a simple electric cell.
Volta discovered that electricity could be created if dissimilar metals were connected by a conductive solution called an electrolyte. This is a simple electric cell. Figure Simple electric cell.

The Electric Battery A battery transforms chemical energy into electrical energy. Chemical reactions within the cell create a potential difference between the terminals by slowly dissolving them. This potential difference can be maintained even if a current is kept flowing, until one or the other terminal is completely dissolved.

The Electric Battery Several cells connected together make a battery, although now we refer to a single cell as a battery as well. Figure (a) Diagram of an ordinary dry cell (like a D-cell or AA). The cylindrical zinc cup is covered on the sides; its flat bottom is the negative terminal. (b) Two dry cells (AA type) connected in series. Note that the positive terminal of one cell pushes against the negative terminal of the other.

Unit of electric current: the ampere, A:
Electric current is the rate of flow of charge through a conductor: The instantaneous current is given by: Unit of electric current: the ampere, A: 1 A = 1 C/s.

Electric Current A complete circuit is one where current can flow all the way around. Note that the schematic drawing doesn’t look much like the physical circuit! Figure (a) A simple electric circuit. (b) Schematic drawing of the same circuit, consisting of a battery, connecting wires (thick gray lines), and a lightbulb or other device.

Electric Current Current is flow of charge.
A steady current of 2.5 A exists in a wire for 4.0 min. (a) How much total charge passed by a given point in the circuit during those 4.0 min? (b) How many electrons would this be? Solution: a. Charge = current x time; convert minutes to seconds. Q = 600 C. b. Divide by electron charge: n = 3.8 x 1021 electrons.

Current Density and Drift Velocity
Electrons in a conductor have large, random speeds just due to their temperature. When a potential difference is applied, the electrons also acquire an average drift velocity, which is generally considerably smaller than the thermal velocity. Figure Electric field E in a uniform wire of cross-sectional area A carrying a current I. The current density j = I/A.

Current Density and Drift Velocity
We define the current density (current per unit area) – this is a convenient concept for relating the microscopic motions of electrons to the macroscopic current: If the current is not uniform: .

Current Density and Drift Velocity
Charges move with a drift velocity along the wire. Total charge within the volume: Time taken to pass through:

Current Density and Drift Velocity
Electron speeds in a wire. A copper wire 3.2 mm in diameter carries a 5.0-A current. Determine (a) the current density in the wire, and (b) the drift velocity of the free electrons. (c) Estimate the rms speed of electrons assuming they behave like an ideal gas at 20°C. Assume that one electron per Cu atom is free to move (the others remain bound to the atom). Solution: a. j = I/A = 6.2 x 105 A/m2 b. The number of free electrons per unit volume is equal to the number of atoms per unit volume. For pure copper, the number of atoms per unit volume can be written as the number of atoms in a mole divided by the volume of a mole of copper; the volume of a mole of copper is given by the atomic mass of copper in grams (63.5 g = 1 mole of copper) divided by the density of copper. This gives 8.4 x 1028 atoms/m3. Then the drift velocity is j/ne = 4.6 x 10-5 m/s. c. If we model the electrons as a three-dimensional single-atom gas, the rms speed is given by the square root of 3kT/m, which is 1.2 x 105 m/s. (This actually underestimates the speed of the electrons by about a factor of 10).

Electric Current How to connect a battery.
What is wrong with each of the schemes shown for lighting a flashlight bulb with a flashlight battery and a single wire? Solution: a. Not a complete circuit. b. Circuit does not include both terminals of battery (so no potential difference, and no current) c. This will work. Just make sure the wire at the top touches only the bulb and not the battery!

Electric Current By convention, current is defined as flowing from + to -. Electrons actually flow in the opposite direction, but not all currents consist of electrons. Figure Conventional current from + to - is equivalent to a negative electron flow from – to +.

Ohm’s Law: Resistance and Resistors
Experimentally, it is found that the current in a wire is proportional to the potential difference between its ends:

Ohm’s Law: Resistance and Resistors
The ratio of voltage to current is called the resistance:

Ohm’s Law In many conductors, the resistance is independent of the voltage; this relationship is called Ohm’s law. Materials that do not follow Ohm’s law are called nonohmic. Figure Graphs of current vs. voltage for (a) a metal conductor which obeys Ohm’s law, and (b) for a nonohmic device, in this case a semiconductor diode. Unit of resistance: the ohm, Ω: 1 Ω = 1 V/A.

Ohm’s Law Current and potential.
Current I enters a resistor R as shown. (a) Is the potential higher at point A or at point B? (b) Is the current greater at point A or at point B? Solution: a. Point A is at higher potential (current flows “downhill”). b. The current is the same – all the charge that flows past A also flows past B.

Ohm’s Law Flashlight bulb resistance.
A small flashlight bulb draws 300 mA from its 1.5-V battery. (a) What is the resistance of the bulb? (b) If the battery becomes weak and the voltage drops to 1.2 V, how would the current change? Figure Flashlight (Example 25–4). Note how the circuit is completed along the side strip. a. R = V/I = 5.0 Ω. b. Assuming the resistance stays the same, the current will drop to 240 mA.

Ohm’s Law Standard resistors are manufactured for use in electric circuits; they are color-coded to indicate their value and precision. Figure The resistance value of a given resistor is written on the exterior, or may be given as a color code as shown above and in the Table: the first two colors represent the first two digits in the value of the resistance, the third color represents the power of ten that it must be multiplied by, and the fourth is the manufactured tolerance. For example, a resistor whose four colors are red, green, yellow, and silver has a resistance of 25 x 104 Ω = 250,000 Ω = 250 kΩ, plus or minus 10%. An alternate example of a simple code is a number such as 104, which means R = 1.0 x 104 Ω.

Ohm’s Law This is the standard resistor color code. Note that the colors from red to violet are in the order they appear in a rainbow.

Ohm’s Law Some clarifications:
Batteries maintain a (nearly) constant potential difference; the current varies. Resistance is a property of a material or device. Current is not a vector but it does have a direction. Current and charge do not get used up. Whatever charge goes in one end of a circuit comes out the other end.

Resistivity The resistance of a wire is directly proportional to its length and inversely proportional to its cross-sectional area: Geometric property The constant ρ, the resistivity, is characteristic of the material.

Resistivity This table gives the resistivity and temperature coefficients of typical conductors, semiconductors, and insulators.

Current Density and Drift Velocity
The electric field inside a current-carrying wire can be found from the relationship between the current, voltage, and resistance. Writing R = ρ l/A, I = jA, and V = El , and substituting in Ohm’s law gives:

Resistivity Speaker wires.
Suppose you want to connect your stereo to remote speakers. (a) If each wire must be 20 m long, what diameter copper wire should you use to keep the resistance less than 0.10 Ω per wire? (b) If the current to each speaker is 4.0 A, what is the potential difference, or voltage drop, across each wire? Solution: a. R = ρl/A; you can solve this for A and then find the diameter. D = 2.1 mm. b. V = IR = 0.40 V.

Resistivity Stretching changes resistance.
Suppose a wire of resistance R could be stretched uniformly until it was twice its original length. What would happen to its resistance? Solution: The total volume of the wire should stay the same; therefore if the length doubles, the cross-sectional area is halved. This increases the resistance by a factor of 4.

Resistivity For any given material, the resistivity increases with temperature: Semiconductors are complex materials, and may have resistivities that decrease with temperature.

Resistivity Resistance thermometer.
The variation in electrical resistance with temperature can be used to make precise temperature measurements. Platinum is commonly used since it is relatively free from corrosive effects and has a high melting point. Suppose at 20.0°C the resistance of a platinum resistance thermometer is Ω. When placed in a particular solution, the resistance is Ω. What is the temperature of this solution? Solution: The resistance is proportional to the resistivity; use the temperature dependence of resistivity to find the temperature. T = 56.0 °C.

Electric Power Power, as in kinematics, is the energy transformed by a device per unit time: or

Electric Power The unit of power is the watt, W.
For ohmic devices, we can make the substitutions:

Calculate the resistance of a 40-W automobile headlight designed for 12 V. Solution: R = V2/P = 3.6 Ω.

Electric Power What you pay for on your electric bill is not power, but energy – the power consumption multiplied by the time. We have been measuring energy in joules, but the electric company measures it in kilowatt-hours, kWh: 1 kWh = (1000 W)(3600 s) = 3.60 x 106 J.

Electric Power Electric heater.
An electric heater draws a steady 15.0 A on a 120-V line. How much power does it require and how much does it cost per month (30 days) if it operates 3.0 h per day and the electric company charges 9.2 cents per kWh? Solution: P = IV = 1800 W W x 3.0 h/day x 30 days = 162 kWh. At 9.2 cents per kWh, this would cost \$15.

Electric Power Lightning bolt.
Lightning is a spectacular example of electric current in a natural phenomenon. There is much variability to lightning bolts, but a typical event can transfer 109 J of energy across a potential difference of perhaps 5 x 107 V during a time interval of about 0.2 s. Use this information to estimate (a) the total amount of charge transferred between cloud and ground, (b) the current in the lightning bolt, and (c) the average power delivered over the 0.2 s. Solution: a. The charge is the change in energy divided by the change in electric potential: Q = 20 C. b. I = Q/t = 100 A. c. P = energy/time = IV = 5 x 109 W = 5 GW.

Power in Household Circuits
The wires used in homes to carry electricity have very low resistance. However, if the current is high enough, the power will increase and the wires can become hot enough to start a fire. To avoid this, we use fuses or circuit breakers, which disconnect when the current goes above a predetermined value.

Power in Household Circuits
Fuses are one-use items – if they blow, the fuse is destroyed and must be replaced. Figure 25-19a. Fuses. When the current exceeds a certain value, the metallic ribbon melts and the circuit opens. Then the fuse must be replaced.

Power in Household Circuits
Will a fuse blow? Determine the total current drawn by all the devices in the circuit shown. Solution: The current is given by I = P/V, where V = 120 V. Adding the currents gives 28.7 A, which exceeds the usual 20-A circuit breakers found in most household applications. The electric heater should probably be on its own circuit.

Superconductivity In general, resistivity decreases as temperature decreases. Some materials, however, have resistivity that falls abruptly to zero at a very low temperature, called the critical temperature, TC. Figure A superconducting material has zero resistivity when its temperature is below TC, its “critical temperature.” At TC, the resistivity jumps to a “normal” nonzero value and increases with temperature as most materials do (Eq. 25–5).

Superconductivity Experiments have shown that currents, once started, can flow through these materials for years without decreasing even without a potential difference. Critical temperatures are low; for many years no material was found to be superconducting above 23 K. Since 1987, new materials have been found that are superconducting below 90 K, and work on higher temperature superconductors is continuing.

Summary A battery is a source of constant potential difference.
Electric current is the rate of flow of electric charge. Conventional current is in the direction that positive charge would flow. Resistance is the ratio of voltage to current:

Summary Ohmic materials have constant resistance, independent of voltage. Resistance is determined by shape and material: ρ is the resistivity.

Summary Power in an electric circuit: Direct current is constant.
Relation between drift speed and current: