Faraday s Law and Inductance


 Paula Wilkerson
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1 Historical Overview Faraday s Law and Inductance So far studied electric fields due to stationary charges and magentic fields due to moving charges. Now study electric field due to a changing magnetic field. Faraday showed that an electric current can be induced in a circuit by a changing magnetic field. These studies lead to Faraday s law of induction. Faraday s law of induction A straight metal wire lies in a uniform magnetic field directed into the page. The wire contains free elctrons. Move the wire with a velocity v to the right. F = qv B implies that a magnetic force acts on the wire. Can use the right hand rule to show that the force on the electrons is downward, along the wire. The electrons move along the wire in response to this force: a current. Consider a loop of wire connected to a sensitive ammeter. Move a magnet toward the loop, the ammeter needle deflects in one direction. If the magnet is stationary, no current is detected. If the magnet is stationary and the coil is moved either away or toward it, a current is detected. An electric force is set up in the coil as long as relative motion occurs between the magnet and the coil. The current in the wire is called an induced current, produced by an induced electromotive force or emf. Faraday concluded that an electric current can be produced by a timevarying magnetic field. In the example above, what causes the timevarying magnetic field? Current cannot be produced by a steady magnetic field. Discuss Figure 23.3 To quantify this, define magnetic flux. 1
2 Consider an element of area da on an arbitrarily shaped open surface. If the magnetic field at the location of this element is B, the magnetic flux through the element is B.dA, where doverlinea is a vector perpendicular to the surface whose magnitude equals area da. Then the total magnetic flux through the surface is Φ B = B.A SI unit of magnetic flux is teslameter squared, where 1 weber or Wb = 1T.m 2. an emf is induced in a circuit when the magnetic flux through the surface bounded by the circuit changes with time. Faradays law of induction states that the emf induced in a circuit is equal to the time rate of change of magnetic flux through the circuit. ɛ = dφ B dt, where Φ B is the magnetic flux through the surface bounded by the circuit. If the circuit is a coil consisting of N identical and concentric loops and if the field lines pass through all loops, the induced emf is Why is cosθ needed here. ɛ = N dφ B ɛ = d dt (BAcosθ). Thus an emf can be induced in a circuit by changing the magnetic flux in several ways: Motional emf magnitude of B can vary with time the area A of the circuit can change with time the angle θ between B and the normal to the plane can change with time any combination of these changes Try QuickQuiz 23.1, go over example An emf is induced in a conductor moving through a magnetic field  motional emf. 2
3 Go over Figure The Alternating Current generator The alternatingcurrent (AC) generator is a device in which energy is transferred in by work and out by electrical transmission: A coil of wire rotated in an external magnetic field by some external agent  this is the work input. For example, in a hydroelectric power plant, falling water directed against the blades of a turbine produces the rotary motion. As the loop rotates, the magnetic flux through it changes with time, inducing an emf and a current in a circuit connected to the coil. Suppose the coil has N turns of the same area A and suppose the coil rotates with a constant angular speed ω about an axis perpendicular to the magnetic field. If θ is the angle between the magnetic field and the direction perpendicular to the plane of the coil, the magnetic flux through the loop at any time t is given by: where θ = ωt. Φ B = BAcosθ = BAcosωt, Thus the induced emf in the coil is ɛ = N dφ B dt = NAB d dt (cosωt) = NABsinωt. Thus the emf varies sinusoidally with time  AC voltage. Lenz s Law Negative sign in Faraday s law. The polarity of the induced emf in a loop is such that it produces a current whose magentic field opposes the change in magnetic flux through the loop. That is the induced current is in a direction such that the induced magnetic field attempts to maintain the original flux through the loop. Go over Figure Induced emfs and Electric Fields A changing magnetic flux indices an emf and a current in a conducting loop. The current is due to an electric field setup by a battery, for example. Can also think of this as the following: a changing magnetic field creates an induced electric field. This electric field applies a force on the charges to cause them to move. 3
4 An electric field is created in a conductor as a result of changing magnetic flux  or  An electric field is always generated by a changing magnetic flux, even in free space where no charges are present. But the induced electric field has different properties to the electrostatic fields produced by stationary charges. Consider a conducting loop of radius r in a uniform magnetic field that is perpendiuclar to the plane of the loop. If the magnetic field changes with time, Faraday s law tells su that ɛ = dφ B /dt is induced in the loop. The induced current thus produced implies an induced electric field, E, that must be tangent to the loop so as to provide an electric force on the charges around the loop. The work done by the electric field on the loop in moving a test charge q once around the loop is W = qɛ. But the force on the charge is qe, and W = F.dr = qe(2πr). Equating these two expressions, we get E = ɛ 2πr. Using this result, together with faraday s law and Φ B = BA, we find E = r 2 db The negative sign indicates that the induced electric field E results in a current that opposes the change in the magnetic field. This result is valid in the absence of a conductor or charges. Thus the general form of Faraday s law of induction is ɛ = E.ds = dφ B The induced electric field E that appears above is a nonconservative field that is generated by a changing magnetic field. Its nonconservative because the work done in moving a charge around a closed path is not zero. Is this like an electrostatic filed? SelfInductance Consider an isolated circuit consisting of a swicth, a resistor and a source of emf. 4
5 When the switch is closed, the current doesnt jump from zero to ɛ/r immediately. As the current increases with time, the magnetic flux through the loop of the circuit itself due to the current also increases with time. This increasing magnetic flux from the circuit induces an emf in the ciruit (back emf) that opposes the change in the net magnetic flux through the loop of the circuit. By Lenz s law, the induced electric field in the wires must therefore be opposite the direction of the current. the opposing emf results in a gradual increase in the emf. The emf setup in this case is called selfinduced emf. From Faraday s law, the induced emf is the negative time rate of change of the magnetic flux. The magnetic flux is proportional to the magnetic field (why?), which in turn is proportional to the current in the circuit (why?). Thus the self induced emf is always proportional to the time rate of change of the current. More quantitatively, for a closely spaced coil of N turns, ɛ L = N dφ B dt = L di dt, where L is a proportionality constant called the inductance of the coil. The inductance of a coil containing N loops is L = NΦ B I. The inductance is the ratio, ɛ L L = di/ The SI unit of inductance is the henry (H): 1H = 1V.s/A. Inductance of a coil depends on geometry. A circuit that contains a coil, such as a solenoid, has a selfinductance that prevents the current from increasing or decreasing instantaneously. RL Circuits A circuit that contains a coil, such as a solenoid, has a selfinductance that prevents the current from increasing or decreasing instantaneously. A circuit element whose main purpose is to provide inductance in a circuit is called an inductor. Consider the circuit shown, consistin gof a resistor, an inductor, a swicth and a battery. 5
6 Ignore the resistance of the battery and suppose the switch is closed at t = 0. The current begins to increase, and due to the opposing current, the inductor produces an emf that opposes the increasing current. The back emf produced by the inductor is ɛ L = L di Applying Kirchoof s loop rule to the circuit: ɛ IR L di dt = 0, where IR is the voltage drop across the resistor. The potential difference across the inductor is given a negative sign because its emf is in the opposite sense to that of the battery. This is a differential equation. Its solution will give I(t). Can show the solution is I(t) = ɛ R (1 e t/τ ), where τ is the time constant of the RL circuit and τ = L/R. Dimension of τ is time. It is the time interval required for the current to reach (1 e 1 ) = of its final value ɛ/r. Note I(t) = 0 when t = 0. And at t = inf, I = ɛ/r. The final current, ɛ/r does not involve the inductance L because the inductor has no effect on the circuit if the current is not changing. Energy Stored in a Magnetic Field Induced emf set up by an inductor prevents a battery from establishing an instantaneous current. Part of the energy supplied by the batter goes into the internal energy in the resistor. The remaining energy is stored in the inductor. From Kirchoffls loop law, we have Iɛ = I 2 R + LI di This says that the rate, Iɛ, at which energy is supplied by the battery equals the rate, I 2 R, at which energy is delivered to the resistor and the rate, LI(dI/dt) at which energy is delivered to the inductor. 6
7 Thus, if U B is the energy stored in the inductor at any time, SOlving this, we get du B dt = LI di U B = 1 2 LI2, where L is constant. This is the energy stored in the magnetic field of the inductor when the current is I. Now consider a solenoid where L = µ 0 n 2 Al. The magnetic field of the solenoid is B = mu 0 ni. Then U B = 1 2 LI2 = 1 2 µ 0n 2 Al( B µ 0 n ) 2 = B2 2µ 0 (Al). Because Al is the volume of the solenoid, the energy stored per unit volume in a magnetic field, or the megnetic energy density, is u B = B2 1µ 0. This formula is valid for any region of space in which a magnetic field exists. Compare to eneergy per unit volume stored in an electric field, 1 2 ɛ 0E 2. 7
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