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Electromagnetism revision Electric fields Chapter 16 Charge is quantized (量子化). Everything is charged in multiples of e, the elementary charge (except quarks!). Charge is conserved (守恒的). The total charge of the universe never changes. Electric field of one point charge q E 1 q ˆ r 2 40 r Coulomb’s Law E 1 q ˆ r 2 40 r + q is positive: the direction of the field is away from the charge. E 1 q ˆ r 2 40 r - q is negative: the direction of the field is towards the charge. Superposition Principle 电场强度叠加原理 + - q2 F4 q4 Enet q4 E1 E2 E3 E1 q1 E3 Enet q3 + Location of q4 E2 Superposition Principle 电场强度叠加原理 In general: Enet E1 E2 E3 or: Enet Ei i The net field at a location in space is the vector sum of the fields contributed by all charged particles at other locations. Dipole 电偶极子 + Hydrogen chloride -q +q d O H + H Water We can define a vector called the dipole moment (电矩). -q p +q d Magnitude: NOTE: This p has nothing to do with momentum! p qd Direction: from the negative (-) charge to the positive (+) charge. 1 p E 40 r 3 Using superposition…. p 1 2p Eaxis 40 r 3 Electric forces on dipoles F F p + E In a uniform field, the net force on a dipole is zero: F F 0 Electric forces on dipoles F F p + E But the net torque around the dipole’s COM is not zero: p E Electric forces on dipoles F F p E This torque will rotate the dipole until it is parallel with the field. Electric forces on dipoles F F p E The dipole-field system has a potential energy: U dipole p E Electric field of distributed charges y y Q r r y Q E x E Uniformly charged rod Uniformly charged ring Eaxis E 2Q L 40 r 1 Q 1 2 40 z Electric field of a uniformly charged plate Q R r r z E Very large plate (R >> z): Q A E 2 0 The field almost doesn’t change with distance, near the plate. Electric potential (电势) Chapter 18 Electric potential (电势) • The potential difference between two points: Vab E ds b a • The potential energy difference for a charge q, moved between two points: • The potential near a point charge, with respect to infinity: U electric qV 1 q1 V (r ) 40 r Electric field is the negative gradient (梯度) of the potential V V V Ex , Ey , Ez x y z V Ex Ex x The potential is like the height of the hill. The field is like the slope of the hill. Just remember: - positive charges go down the hill - negative charges go up! Potential along the axis of a ring dQ R Potential obeys the superposition principle, just like the field. x2 R2 x Potential due to one small piece: dV 1 dQ 40 x2 R2 Potential along the axis of a ring dQ Potential obeys the superposition principle, just like the field. x2 R2 R x Integrate: V dV 1 1 40 x R 2 2 dQ Potential along the axis of a ring dQ Potential obeys the superposition principle, just like the field. x2 R2 R x Integrate: V 1 Q 40 x2 R2 Field along the axis of a ring The strength of the field is the negative of the potential gradient: E V Ex x 1 40 x x 2 R 2 Q Field along the axis of a ring The strength of the field is the negative of the potential gradient: E Ex 1 Qx 40 x R 2 2 3/ 2 Field along the axis of a ring We already calculated this field the hard way. It is often easier to first calculate the potential, then use its gradient to get the field. E Ex 1 Qx 40 x R 2 2 3/ 2 Potential in a conductor B At equilibrium, the field inside the conductor must be zero. A + VAB 0 Potential in a conductor So the potential inside a conductor (and at the surface) must be constant. + V constant Capacitors and dielectrics Chapter 19 Two uniformly charged plates: A capacitor (电容器) R d - + + + + + + + Einside Q A 0 There is also a small field outside the plates: Efringe Q A d 2 0 R Definition of capacitance Q C V + + + + + + + - Capacitance C measures: - how much charge Q we must put on the plates - to achieve a certain potential difference ΔV. For parallel-plate capacitor: A = area of plates 0 A C d Units: the Farad (F) d = separation of plates Fby you + + + + + + + Energy density in an electric field Eother plate Fby plate - 1 2 Energy / v olume 0 E 2 d It is true for any electric field, not just in a capacitor! Electric fields contain energy. (Also momentum!) I insert a dielectric between the plates. What happens? +Q + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + + + + + + + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + + - + - + - + - + Epolar - + Eplates - -Q - Etotal Polarization of the dielectric reduces the net electric field between the plates. Eplates The dielectric constant ε depends on the material. +Q + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + + + + + + + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + Etotal - -Q - It also reduces the potential difference between the plates. V Vvacuum The dielectric constant ε depends on the material. +Q + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + + + + + + + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + Etotal - -Q - Magnetic fields from currents Chapter 21 Magnetic field of a single moving charge: The Biot-Savart Law B r̂ + v “Permeability of free space” 0 qv rˆ B 2 4 r Units: tesla (T) 0 tesla m 7 110 exactly 4 coulomb / s For a very long wire (L >> r): 0 2i B 4 R Far from a current loop (z >> R) 0 iA B 4 z 3 Btotal Current loops are magnetic dipoles. Baxis iA The magnetic dipole moment 0 4 z 3 Magnetic field inside a solenoid 螺线管 . Binside 0 ni where n is the number of loops per unit length. Magnetic forces Chapter 20 Moving electric charges make magnetic fields… and magnetic fields make forces on moving electric charges. Fmagnetic qv B F ( e)v B - v vB The force is always perpendicular to the direction of motion: it cannot change the particle’s speed. Velocity selector: B into page FB + FE E qE qvB E v B Magnetic force on a current-carrying wire i L F i dl B B Magnetic force on a current-carrying wire i F into page L B F iL B (uniform field) Torque on a current loop B B b i a B i out Magnetic dipole moment a ibB sin 2 B i in a ibB sin 2 B i out i in B A magnetic dipole will align with the magnetic field. Gauss’ Law Chapter 17 r E L q E d A 0 Uniformly charged spherical shell (outside) E dA q r Uniformly charged spherical shell (inside) q E r Excess charge in a conductor is always on the surface. q E dA Q E 0 0 0 0 Gaussian surface Net charge is zero inside a conductor. Gauss’ Law for magnetic fields B d A 0 Ampere’s Law Chapter 21 B d s i 0 enc Magnetic field inside a cylindrical current-carrying conductor B “Amperian loop” i r R B d s B 2 r r ienclosed i 2 R 2 Magnetic field inside a cylindrical current-carrying conductor B “Amperian loop” i r R 0i B r 2 2R (direction from right-hand rule!) Faraday’s Law Chapter 23 ENC d B EMF dt ENC i increasing Which direction does the electric field curl? dB Right thumb along dt ENC Fingers curl in direction of ENC dB dt i increasing i2 ENC can induce current in a wire around the changing magnetic field. d B EMF dt EMF i2 resistance i1 Example Ammeter 电表 What current will the ammeter measure? Wire R 0.5 The magnetic field in the solenoid increases from 0.1 T to 0.7 T in 0.2 seconds. B B BAsolenoid Solenoid Asolenoid 3 cm 2 d B B EMF Asolenoid dt t EMF i 1.8 10 3 A R Motional EMF (动生电动势) vt B L i i d B EMF dt v d B EMF dt This equation works, no matter if • the magnetic field is changing (Faraday’s Law), • the area of the circuit is changing (motional EMF), • or both! Maxwell’s Equations Chapter 21, 24 B ds 0 B ds 0i Something missing here… Ampere’s Law is incomplete. d E i 0 dt How Maxwell fixed Ampere’s Law: B ds 0ienclosed d E i 0 dt How Maxwell fixed Ampere’s Law: d E B ds 0 ienclosed 0 dt Maxwell’s Equations qinside E dA Gauss’ Law for electric fields B dA 0 Gauss’ Law for magnetic fields 0 d E ds dt B dA Faraday’s Law d B ds 0 ienclosed 0 dt E dA Ampere-Maxwell Law Maxwell’s equations predict travelling waves. v v 1 0 0 Maxwell’s equations predict light. v v 3.0 10 m/s 8