doPhysics · Electromagnetism 1
Electric Current and Resistance
Current as charge per second and its density, the drift of free electrons, Ohm's law and resistivity, how resistance changes with temperature, non-ohmic conductors, Joule heating and electric power, and bulbs in series and in parallel.
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Why this chapter
Why this chapter exists
Until now every charge in this unit has stood still. Let charge flow and a new world opens: wires, heaters, bulbs, fuses and the electricity bill. This chapter looks inside a copper wire. Free electrons dart about at great speed and yet drift along it at a fraction of a millimetre per second. From that slow drift come Ohm's law, the resistance of every wire and the heat of every heater. The chapter ends with the question every household meets: which bulb glows brighter, and why.
You will learn
By the end of this chapter you will
- Define current and current density, and find the charge a changing current carries from the area under its graph.
- Explain the drift of free electrons, derive \(v_d = eE\tau/m\) and \(I = neAv_d\), and explain why a bulb lights at once although the drift is slow.
- State Ohm's law, use \(R = \rho L/A\) for wires and blocks, derive \(J = \sigma E\) with \(\rho = m/ne^2\tau\), and find the resistance of a stretched wire.
- Use \(R = R_0(1 + \alpha\Delta T)\), explain why metals, semiconductors and alloys differ, and read the resistance of a non-ohmic conductor from its graph.
- Derive \(H = I^2Rt\), choose the right form of \(P = VI = I^2R = V^2/R\), and work out electricity bills and the efficiency of heaters and motors.
- Find the power and brightness of rated bulbs in series and in parallel, and the ratings of fuses, coils and resistors.
Before you begin
Before you begin
You should know the electric field as force per unit charge, from the first chapter of this unit. You should know the potential difference as work per coulomb, from the second. You should be able to use \(R_t = R_0(1 + \alpha t)\) from the platinum thermometer, and \(Q = mc\Delta T\) and latent heat from Calorimetry and Change of State. You should be able to find the area under a graph by integration.
Lessons
1
Electric Current
A current is charge crossing a section each second, \(I = q/t\), in amperes. Its direction is the way positive charge would move, against the electrons in a metal. A current that changes carries the area under its i-t graph, and the current density \(J = I/A\) tells how crowded the flow is.
Current as charge per secondCurrent that changes with timeCurrent density
2
Drift Velocity
Free electrons dart about at \(10^5\,\text{m/s}\) and carry no current. A field adds a slow drift, \(v_d = eE\tau/m\), restarted at every collision. The current is \(I = neAv_d\), so the drift is a fraction of a millimetre per second, the bulb still lights at once, and a narrower wire needs a faster drift.
Free electrons and driftCurrent from the drift speedWhy the bulb lights at onceDrift through a narrowing wire
3
Ohm's Law and Resistivity
For a metal at a fixed temperature the current is proportional to the potential difference, \(V = IR\). The resistance \(R = \rho L/A\) splits into a property of the material and the shape of the sample; inside the metal the same law reads \(J = \sigma E\) with \(\rho = m/ne^2\tau\). Stretching a wire makes \(R\) grow as the square of its length.
Ohms lawResistivity and the shape of a wireOhms law inside the metalStretching a wire
4
Resistance and Temperature, Non-Ohmic Conductors
A metal resists more when hot, \(R = R_0(1 + \alpha\Delta T)\), because its ions shake harder and cut the relaxation time. A semiconductor resists less, because heat frees more carriers, and some alloys hardly change. A lamp filament heats as its current grows, so its I-V graph bends and it has a resistance only at each current.
Resistance rises with temperature in metalsSemiconductors and alloysNon ohmic conductors
5
Joule Heating and Electric Power
Charge falling through a potential difference in a resistor hands its energy to the ions as heat, \(H = I^2 Rt\). The power \(P = VI = I^2 R = V^2/R\), and the right form depends on what is held fixed. Electricity is sold in kilowatt-hours, and a heater's efficiency is the share of its energy that reaches the water.
Heating effect of a currentPower and ratingsThe electricity bill in unitsKettles immersion rods and efficiency
6
Bulbs in Series and in Parallel
A bulb's rating fixes its resistance, \(R = V^2/P\), so the lower-rated bulb has the larger resistance. In series the same current flows and that bulb glows brighter; in parallel each bulb has the full voltage and glows as rated. The same reasoning times a kettle's coils, blows a fuse and sets the safe voltage of a resistor.
Resistance of a rated bulbBulbs in seriesBulbs in parallelCoils fuses and safe ratings
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%The Workbook
Questions to solve on paper, marked by you.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- ampere
- The SI unit of current, A, a base unit of SI. One ampere carries one coulomb per second.
- conductance
- The reciprocal of resistance, \(G = 1/R\), measured in siemens, S.
- conductivity
- The reciprocal of resistivity, \(\sigma = 1/\rho\), in S/m. It links the current density to the field in a material, \(J = \sigma E\).
- conventional current
- The direction of current taken as the direction in which positive charge moves. In a metal it points against the motion of the free electrons.
- current density
- The current per unit area of a section drawn at right angles to the flow, \(J = I/A\), in A/m². A vector along the conventional current.
- drift velocity
- The small average velocity of the free electrons in a conductor carrying a current, \(v_d = eE\tau/m\), directed against the field.
- efficiency
- The fraction of the energy supplied to a device that goes to the useful purpose, such as heating the water or turning the shaft.
- electric current
- The net charge that crosses a section of a conductor per second, \(I = \Delta q/\Delta t\), in amperes. A scalar with a direction along the conductor.
- electrolyte
- A liquid that conducts by the motion of positive and negative ions, such as salt water or a solution of copper sulphate.
- free electron
- An electron in a metal that is bound to no single atom and moves through the whole lattice. Copper has about \(8.5 \times 10^{28}\) of them per cubic metre.
- fuse
- A short wire in the main line of a circuit that melts and breaks the circuit when the current passes its rated value.
- Joule heating
- The heating of a conductor by a current, \(H = I^2Rt\): the electrons hand the energy the field gives them to the ions at each collision.
- kilowatt-hour
- The unit of electrical energy on a bill, the energy a 1 kW device uses in one hour: \(1\,\text{kWh} = 3.6 \times 10^6\,\text{J}\).
- mobility
- The drift speed of a charge carrier per unit field, \(\mu = v_d/E = e\tau/m\), in m²/(V s).
- non-ohmic conductor
- A conductor whose V-I graph is not a straight line through the origin, such as a lamp filament or a diode. Its \(V/I\) depends on the current.
- ohm
- The SI unit of resistance, Ω. A conductor has one ohm if one volt across it drives one ampere through it.
- Ohm's law
- The law that the current through a metallic conductor at a fixed temperature is proportional to the potential difference across it, \(V = IR\) with \(R\) constant.
- ohmic conductor
- A conductor that obeys Ohm's law: its V-I graph is a straight line through the origin, and \(V/I\) is the same at every current.
- parallel
- An arrangement in which devices are joined across the same two points, so each has the same potential difference.
- relaxation time
- The average time since the last collision of the free electrons in a metal, \(\tau\). For copper it is about \(2.5 \times 10^{-14}\,\text{s}\).
- resistance
- The ratio of the potential difference across a conductor to the current through it, \(R = V/I\), in ohms.
- resistivity
- The resistance of a unit cube of a material, \(\rho\) in \(R = \rho L/A\), in Ω m. A property of the material, not of the sample; \(\rho = m/ne^2\tau\).
- series
- An arrangement in which devices sit one after the other on a single path, so the same current flows through each.
- steady current
- A current that does not change with time. It carries the charge \(q = It\) in a time \(t\).
- temperature coefficient of resistance
- The fractional change of resistance per degree, \(\alpha\) in \(R = R_0(1 + \alpha\Delta T)\). Positive for metals, negative for semiconductors, nearly zero for some alloys.
- thermistor
- A small bead of semiconductor whose resistance falls steeply as it warms, used to sense temperature.