Chapter 21 Magnetic Induction Lecture 12 21.1 Why is it called Electromagnetism? 21.2 Magnetic Flux and Faraday s Law 21.3 Lenz s Law and Work-Energy Principles 21.4 Inductance 21.5 RL Circuits 21.6 Energy Stored in a Magnetic Field
Electromagnetism and Magnetic Induction Electric and magnetic phenomena were connected by Ørsted in 1820 He discovered an electric current in a wire can exert a force on a compass needle Indicated a electric field can lead to a force on a magnet He concluded an electric field can produce a magnetic field Did a magnetic field produce an electric field? Yes! Experiments were done by Michael Faraday This effect is called magnetic induction This links electricity and magnetism in a fundamental way Section 21.1
Faraday s Experiment Faraday attempted to observe an induced electric field He used an ammeter instead of a light bulb If the bar magnet was in motion, a current was observed If the magnet is stationary, the current and the electric field are both zero Section 21.1
Another Faraday Experiment A solenoid is positioned near a loop of wire with the light bulb He passed a current through the solenoid by connecting it to a battery When the current through the solenoid is constant, there is no current in the wire When the switch is opened or closed, the bulb does light up Section 21.1
Conclusions from Experiments An electric current is produced during those instances when the current through the solenoid is changing Faraday s experiments show that an electric current is produced in the wire loop only when the magnetic field at the loop is changing A changing magnetic field produces an electric field An electric field produced in this way is called an induced electric field The phenomena is called electromagnetic induction Section 21.1
Magnetic Flux Faraday developed a quantitative theory of induction now called Faraday s Law The law shows how to calculate the induced electric field in different situations Faraday s Law uses the concept of magnetic flux Magnetic flux is similar to the concept of electric flux Let A be an area of a surface with a magnetic field passing through it The flux is Φ B = B A cos θ Section 21.2
Magnetic Flux, cont. If the field is perpendicular to the surface, Φ B = B A If the field makes an angle θ with the normal to the surface, Φ B = B A cos θ If the field is parallel to the surface, Φ B = 0 The SI unit of magnetic flux is the Weber (Wb) 1 Wb = 1 T. m 2 Section 21.2
Faraday s Law Faraday s Law indicates how to calculate the potential difference that produces the induced current Written in terms of the electromotive force induced in the wire loop t Φ B = B A cos θ The magnitude of the induced emf equals the rate of change of the magnetic flux The negative sign is Lenz s Law B Section 21.2
Applying Faraday s Law The ε is the induced emf in the wire loop Its value will be indicated on the voltmeter It is related to the electric field directly along and inside the wire loop The induced potential difference produces the current
Flux Though a Changing Area A magnetic field is constant and in a direction perpendicular to the plane of the rails and the bar Assume the bar moves at a constant speed The magnitude of the induced emf is ε = B L v Φ B = B A cos θ t The current leads to power dissipation in the circuit (by the resistor) Section 21.2 B
Conservation of Energy The mechanical power put into the bar by the external agent is equal to the electrical power delivered to the resistor Energy is converted from mechanical to electrical, but the total energy remains the same Conservation of energy is obeyed by electromagnetic phenomena Section 21.2
Electrical Generator Need to make the rate of change of the flux large enough to give a useful emf t B Φ B = B A cos θ Use rotational motion instead of linear motion A permanent magnet produces a constant magnetic field in the region between its poles Section 21.2
Changing a Magnetic Flux, Summary Section 21.2
Lenz s Law Lenz s Law gives an easy way to determine the sign of the induced emf Lenz s Law states the magnetic field produced by an induced current always opposes any changes in the magnetic flux Φ B = B A cos θ t B Section 21.3
Lenz s Law, Example 1 Assume a metal loop in which the magnetic field passes upward through it Assume the magnetic flux increases with time The magnetic field produced by the induced emf must oppose the change in flux Therefore, the induced magnetic field must be downward and the induced current will be clockwise Section 21.3
Lenz s Law, Example 2 Assume a metal loop in which the magnetic field passes upward through it Assume the magnetic flux decreases with time The magnetic field produced by the induced emf must oppose the change in flux Therefore, the induced magnetic field must be upward and the induced current will be counterclockwise Section 21.3
Inductance In some cases, you must include the induced flux When the switch is closed, a sudden change in current occurs in the coil This current produces a magnetic field An emf and current are induced in the coil t B Section 21.4
Inductor A coil is type of circuit element called an inductor Many inductors are constructed as small solenoids Almost any coil or loop will act as an inductor Whenever the current through an inductor changes, a voltage is induced in the inductor that opposes this change This phenomenon is called self-inductance The current changing through a coil induces a current in the same coil The induced current opposes the original applied current, from Lenz s Law Section 21.4
Inductance of a Solenoid Faraday s Law can be used to find the inductance of a solenoid N B BAcos, B o I o L is the symbol for inductance L NA 2 l The voltage across the solenoid can be expressed in terms of the inductance I L V t emf,l d N B d Li L di dt dt dt The results apply to all coils or loops of wire The value of L depends on the physical size and shape of the circuit element The voltage drop across an inductor is The unit of inductance is the Henry 1 H = 1 V. s / A I VL L t t Section 21.4 B
Mutual Inductance It is possible for the magnetic field of one coil to produce an induced current in a second coil The coils are connected indirectly through the magnetic flux The effect is called mutual inductance Section 21.4
RL Circuit DC circuits may contain resistors, inductors, and capacitors The voltage source is a battery or some other source that provides a constant voltage across its output terminals Behavior of DC circuits with inductors Immediately after any switch is closed or opened, the induced emfs keep the current through all inductors equal to the values they had the instant before the switch was thrown After a switch has been closed or opened for a very long time, the induced emfs are zero Section 21.5
RL Circuit Example t B Section 21.5
RL Circuit Example, Analysis The presence of resistors and an inductor make the circuit an RL circuit The current starts at zero since the switch has been open for a very long time At t = 0, the switch is closed, inducing a potential across the inductor Just after t = 0, the current in the second loop is zero After the switch has been closed for a long time, the voltage across the inductor is zero Section 21.5
Time Constant for RL Circuit The current at time t is found by V t I e R 1 is called the time constant of the circuit For a single resistor in series with a single inductor, L R The voltage is given by VL Ve t Section 21.5
Real Inductors Most practical inductors are constructed by wrapping a wire coil around a magnetic material Filling a coil with magnetic material greatly increases the magnetic flux through the coil and therefore increases the induced emf The presence of magnetic material increases the inductance Most inductors contain a magnetic material inside which produces a larger value of L in a smaller package Section 21.5
Energy in an Inductor Energy is stored in the magnetic field of an inductor The energy stored in an inductor is PE ind = ½ L I 2 Very similar in form to the energy stored in the electric field of a capacitor The expression for energy can also be stated as PE ind 1 NA I 2 2 o In terms of the magnetic field, 1 PEmag B 2 volume 2 o 2 Section 21.6
Energy in an Inductor, cont. Energy contained in the magnetic field actually exists anywhere there is a magnetic field, not just in a solenoid Can exist in empty space The potential energy can also be expressed in terms of the energy density in the magnetic field PEmag 1 2 energy density umag B volume 2 o This expression is similar to the energy density contained in an electric field Section 21.6
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