Motional emf, final For equilibrium, qe = qvb or E = vb A potential difference is maintained between the ends of the conductor as long as the conductor continues to move through the uniform magnetic field If the direction of the motion is reversed, the polarity of the potential difference is also reversed
Sliding Conducting Bar A bar moving through a uniform field and the equivalent circuit diagram Assume the bar has zero resistance The work done by the applied force appears as internal energy in the resistor R
Induced emf and Electric Fields An electric field is created in the conductor as a result of the changing magnetic flux Even in the absence of a conducting loop, a changing magnetic field will generate an electric field in empty space This induced electric field is nonconservative Unlike the electric field produced by stationary charges
Induced emf and Electric Fields, cont. The emf for any closed path can be expressed as the line integral of E. ds over the path Faraday s law can be written in a general form: E d s Ñ = dφ dt B
Generators Electric generators take in energy by work and transfer it out by electrical transmission The AC generator consists of a loop of wire rotated by some external means in a magnetic field
Rotating Loop Assume a loop with N turns, all of the same area rotating in a magnetic field The flux through the loop at any time t is Φ B = BA cos θ = BA cos ωt
Induced emf in a Rotating Loop The induced emf in the loop is dφb ε = N dt = NABωsinωt This is sinusoidal, with ε max = NABω
Active Figure 31.21 (SLIDESHOW MODE ONLY)
Induced emf in a Rotating Loop, cont. ε max occurs when ωt = 90 o or 270 o This occurs when the magnetic field is in the plane of the coil and the time rate of change of flux is a maximum ε = 0 when ωt = 0 o or 180 o This occurs when B is perpendicular to the plane of the coil and the time rate of change of flux is zero
DC Generators The DC (direct current) generator has essentially the same components as the AC generator The main difference is that the contacts to the rotating loop are made using a split ring called a commutator
DC Generators, cont. In this configuration, the output voltage always has the same polarity It also pulsates with time To obtain a steady DC current, commercial generators use many coils and commutators distributed so the pulses are out of phase
Active Figure 31.23 (SLIDESHOW MODE ONLY)
Motors Motors are devices into which energy is transferred by electrical transmission while energy is transferred out by work A motor is a generator operating in reverse A current is supplied to the coil by a battery and the torque acting on the current-carrying coil causes it to rotate
Motors, cont. Useful mechanical work can be done by attaching the rotating coil to some external device However, as the coil rotates in a magnetic field, an emf is induced This induced emf always acts to reduce the current in the coil The back emf increases in magnitude as the rotational speed of the coil increases
Eddy Currents Circulating currents called eddy currents are induced in bulk pieces of metal moving through a magnetic field The eddy currents are in opposite directions as the plate enters or leaves the field Eddy currents are often undesirable because they represent a transformation of mechanical energy into internal energy
Maxwell s Equations, Introduction Maxwell s equations are regarded as the basis of all electrical and magnetic phenomena Maxwell s equations represent the laws of electricity and magnetism that have already been discussed, but they have additional important consequences
Maxwell s Equations, Statement Ñ S Ñ S Ñ Ñ E da= B da= E ds q ε Gauss's law ( electric) 0 Gauss's law in magnetism B ds= μ I + ε μ o dφ = dt B o o o Faraday's law dφ dt E Ampere-Maxwell law
Maxwell s Equations, Details Gauss s law (electrical): E d A = Ñ The total electric flux through any closed surface equals the net charge inside that surface divided by ε o This relates an electric field to the charge distribution that creates it S q ε o
Maxwell s Equations, Details 2 Gauss s law (magnetism): B d A = Ñ The total magnetic flux through any closed surface is zero This says the number of field lines that enter a closed volume must equal the number that leave that volume This implies the magnetic field lines cannot begin or end at any point Isolated magnetic monopoles have not been observed in nature S 0
Maxwell s Equations, Details 3 dφ Faraday s law of Induction: Ñ E B d s = dt This describes the creation of an electric field by a changing magnetic flux The law states that the emf, which is the line integral of the electric field around any closed path, equals the rate of change of the magnetic flux through any surface bounded by that path One consequence is the current induced in a conducting loop placed in a time-varying B
Maxwell s Equations, Details 4 The Ampere-Maxwell law is a generalization of Ampere s law B d s = μ I + ε μ Ñ o o o dφ dt It describes the creation of a magnetic field by an electric field and electric currents The line integral of the magnetic field around any closed path is the given sum E
The Lorentz Force Law Once the electric and magnetic fields are known at some point in space, the force acting on a particle of charge q can be calculated F = qe + qv x B This relationship is called the Lorentz force law Maxwell s equations, together with this force law, completely describe all classical electromagnetic interactions
Chapter 32 Inductance
Some Terminology Use emf and current when they are caused by batteries or other sources Use induced emf and induced current when they are caused by changing magnetic fields When dealing with problems in electromagnetism, it is important to distinguish between the two situations
Self-Inductance When the switch is closed, the current does not immediately reach its maximum value Faraday s law can be used to describe the effect
Self-Inductance, 2 As the current increases with time, the magnetic flux through the circuit loop due to this current also increases with time This corresponding flux due to this current also increases This increasing flux creates an induced emf in the circuit
Self-Inductance, 3 The direction of the induced emf is such that it would cause an induced current in the loop which would establish a magnetic field opposing the change in the original magnetic field The direction of the induced emf is opposite the direction of the emf of the battery This results in a gradual increase in the current to its final equilibrium value
Self-Inductance, 4 This effect is called self-inductance Because the changing flux through the circuit and the resultant induced emf arise from the circuit itself The emf ε L is called a self-induced emf
Self-Inductance, Coil Example A current in the coil produces a magnetic field directed toward the left (a) If the current increases, the increasing flux creates an induced emf of the polarity shown (b) The polarity of the induced emf reverses if the current decreases (c)
Self-Inductance, Equations A induced emf is always proportional to the time rate of change of the current d I εl = L dt L is a constant of proportionality called the inductance of the coil and it depends on the geometry of the coil and other physical characteristics
Inductance of a Coil A closely spaced coil of N turns carrying current I has an inductance of NΦB L = = I d The inductance is a measure of the opposition to a change in current ε I L dt
Inductance Units The SI unit of inductance is the henry (H) V s 1H = 1 A Named for Joseph Henry (pictured here)
Inductance of a Solenoid Assume a uniformly wound solenoid having N turns and length l Assume l is much greater than the radius of the solenoid The interior magnetic field is B = μ n I = μ o o N l I
Inductance of a Solenoid, cont The magnetic flux through each turn is Φ = BA = B Therefore, the inductance is L NΦB = = I μ This shows that L depends on the geometry of the object o NA l 2 μona l I
RL Circuit, Introduction A circuit element that has a large selfinductance is called an inductor The circuit symbol is We assume the self-inductance of the rest of the circuit is negligible compared to the inductor However, even without a coil, a circuit will have some self-inductance