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Charge and Current
- Electric current is a base quantity defined in terms of the force between two long, parallel, current-carrying conductors.
- Current I is rate of flow of charge. I = ΔQ/(Δt ΔQ) = I Δt
- Current is measured using an ammeter connected in series.
- 1 coulomb (C) is the quantity of charge which passes when a current of 1 ampere passes for a time of 1 second.
- The elementary charge is e = 1.60 × 10−19 C. This is the size of the negative charge of an electron and the positive charge of a proton.
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Charge Carriers
- In a metal, charge is carried by movement of free electrons; in an electrolyte, charge is carried by positive and negative ions.
- The conventional current is the direction in which positive charges would move. The electron current is in the opposite direction.
- Conductors, semiconductors and insulators have different numbers n of charge carriers per unit volume.
- In a metal wire of cross-sectional area A with electrons moving with drift velocity v, the current is I = nAve
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Potential Difference and E.m.f
- The potential difference between two points is defined as the work done in moving unit charge between those points. V = W/Q
- The p.d. or voltage across a component is measured using a voltmeter (of very high resistance) connected across it, i.e. in parallel with it.
- There is a p.d. of 1 volt between two points if 1 joule of work is done when 1 coulomb of charge moves between the points. 1 V = 1 JC−1
- The electromotive force (e.m.f.) of a source is defined as the work done in moving unit charge round a circuit
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Ressistance
- The functions of a resistor in a circuit are to control the current, and to dissipate energy in heating.
- The resistance R of a conductor = (p.d. V across the conductor)/(current through it). R = V/I
- A conductor has a resistance of 1 ohm (Ω) if the current through it is 1 ampere when the p.d. across it is 1 volt.
- Ohm’s law: for a metallic conductor at constant temperature, the current through it is proportional to the p.d. across it.
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Current-Voltage Characteristics
- The I–V characteristic (graph) for a fixed resistor at constant temperature is a straight line through the origin.
- The I–V characteristic for a filament lamp curves away from the current axis, because the resistance of the lamp increases with temperature.
- The I–V characteristic for a diode shows zero current for negative p.d.s, and for positive p.d.s up to about 0.5 V. The current then rises steeply, showing that the diode has low resistance in theforward direction.
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Resistivity
- The resistivity of a conductor of length L, cross-sectional area A and resistance R is ρ = RA/L. Thus R = ρL/A
- The unit of resistivity is Ωm. The resistivity of a metallic alloy used to make resistance wires is, typically, ρ = 4.9 × 10−7 Ωm.
- The resistance of a metal increases with temperature, because the increased vibration of atoms impedes the movement of electrons.
- The resistance of an (NTC) thermistor decreases with temperature because the greater energy releases more charge carriers.
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Power in Electrical Circuits
- Power P developed in a
- circuit is P = VI = I2R = V2/R
- Energy transferred = power × time W = Pt = ItV
- Fuse wire is made of tinned copper, which has a low melting point. If the current rises too high, the fuse melts and protects the device.
- The kilowatt-hour is the energy transferred by a device of power 1 kW in a time of 1 hour. Electricity companies use the kWh for costing. 1 kWh = 1 kilowatt × 1 hour = 1000 J s−1 × 3600 s = 3.6 MJ
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Series and Parallel Circuits
- Kirchhoff’s first law: at any circuit junction, total current in = total current out. This is a consequence of the conservation of charge. It follows that the current is constant round any series circuit.
- Kirchhoff’s second law: round any circuit loop, the sum of the e.m.f.s is equal to the sum of the p.d.s.
- Σ(e.m.f.) = Σ(IR). This is a consequence of the conservation
- of energy.
- Resistors or circuit components connected in parallel have the same potential difference across them
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Resistors in Circuits
- For three resistors in series: Rs = R1 + R2 + R3
- For three resistors in parallel: 1/Rp = 1/R1 + 1/R2 + 1/R3
- A cell has internal resistance due tothe resistance of its chemicals. Some energy is dissipated in the internal resistance, and the terminal p.d. of a cell or battery which delivers a current is less than the e.m.f.
- For a cell of e.m.f. E and internal resistance r passing a current I through an external resistor R, E = I(R + r) The terminal p.d. is therefore V = IR = E − Ir
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Potential Dividers
- A p.d. V is applied across two resistors R1 and R2 in series. The p.d.s across the resistors are V1 = VR1/(R1 + R2) and V2 = VR2/(R1 + R2)
- The resistance of a light-dependent resistor (LDR) decreases as the intensity of light falling on it increases.
- The resistance of an NTC thermistor decreases with increasing temperature.
- LDRs and thermistors are used as sensors in potential divider circuits to give p.d.s which change with light or temperature
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Wave motion
- A progressive wave transfers energy from one place to another, but the medium only vibrates or oscillates.
- All wave motions, including water waves, sound, light and microwaves can show reflection, refraction, diffraction and interference.
- In transverse waves the vibration of the medium is perpendicular to the wave direction. Examples: waves on water, all electromagnetic waves.
- In longitudinal waves the vibration of the medium is along the wave direction. Examples: waves in a slinky spring, sound.
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Wave Measurements
- The amplitude A of a wave motion is the maximum displacement of the vibrating medium, or the maximum value of an oscillating signal.
- The wavelength λ of a wave motion is the distance between successive points in phase.
- The frequency f of a wave motion is the number of oscillations passing a point per unit time. Frequency is measured in hertz (Hz). The period T is the time duration for one complete oscillation. T = 1/f
- For all wave motions: speed = frequency × wavelength v = fλ
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Electromagnetic Waves
- All electromagnetic waves travel at the same speed c in a vacuum or free space. c = 3.00 × 108 m s−1
- All e-m waves are transverse waves and can be polarised.
- The wavelengths of visible light range approximately from 400 nm for violet to 700 nm for red.
- Typical wavelengths for other e-m radiations are: gamma X-rays ultraviolet infrared microwaves radio
- 10−12 m 10−10 m 10−8 m 10−5 m 10−2 m 1–1000 m
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Polarisation
- In unpolarised waves the vibration of the medium is in all possible directions in the plane perpendicular to the direction of wave travel.
- In plane-polarised waves the vibration of the medium is only in one plane which is parallel to the direction of wave travel.
- Polarisation occurs with transverse waves but not longitudinal waves. Light reflected from a non-metallic surface is partially polarised.
- Malus’ law: I = Io cos2 θ where I = transmitted intensity, Io = incident intensity, θ = angle between planes of incident radiation and medium
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Superposition of Waves and diffraction
- The Principle of Superposition: if twowave motions are superposed, the resulting displacement at any point is the vector sum of the individual displacements.
- At any point: intensity ∝ (amplitude of wave motion)2
- Diffraction is the spreading out of waves when they pass through an aperture or round an obstacle.
- If light of wavelength λ is incident on a diffraction grating of spacing d, then a pattern of order n is seen at angle θ where nλ = d sin θ
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Interference
- Two coherent sources have the same frequency and a constant phase difference between them.
- Constructive interference occurs when two wave trains arrive at a point in phase and reinforce. Path difference from the sources = nλ
- Destructive interference occurs when two wave trains arrive at a point with phase difference 180° and cancel. Path difference = (n + ½)λ
- For light of wavelength λ incident on double slits of spacing a, the fringe separation on a screen at distance D is x = λD/a
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Stationary Waves
- Stationary (standing) waves are formed when two identical waves travelling in opposite directions are superposed.
- Nodes are formed where the amplitude of oscillation is zero. Antinodes are formed where the amplitude of oscillation is maximum.
- The separation of adjacent nodes = λ/2
- The fundamental mode of oscillation is the lowest frequency f0, when there is one antinode, and length = λ/2. Frequencies which are integral multiples of f0 give harmonics, with more loops in the pattern.
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Photons
- A photon is one quantum of energy of electromagnetic radiation.
- The energy of a photon of radiation of frequency f is E = hf = hc/λ where h is the Planck constant: h = 6.63 × 10−34 J s
- When an electron is accelerated from rest through a p.d. V, the kinetic energy is ½mv2 = eV
- The Planck constant h may be estimated using the voltage required to excite an LED: hc/λ = eV
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The photoelectric effect
- A negatively charged and freshlycleaned zinc plate will lose electrons if ultraviolet radiation is incident on it.
- Photoelectric effect shows that e-m radiation has particle properties.
- The work function Φ of a metal is the minimum energy to release an electron from its surface. The threshold frequency is the minimum frequency of radiation to release electrons from the metal’s surface.
- One photon of frequency f may release one electron. The maximum KE of the electron released is ½mvmax2 = hf − Φ
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Wave-particle duality, spectra
- The diffraction of electrons by graphite shows that they have wave properties. Diffraction of electrons can show the size of a nucleus.
- De Broglie equation: The wavelength of the wave motion associated with a moving particle of momentum mv is λ = h/mv
- The line spectrum from a gaseous-discharge lamp provides evidence of discrete energy levels for the electrons in an atom.
- If an electron falls from energy level E1 to energy level E2, then a photon of frequency f is emitted, where hf = E1 − E2
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