Chapter 21 Magnetic Induction

Electric and magnetic forces both act only on particles carrying an electric charge Moving electric charges create a magnetic field A changing magnetic field created an electric field This effect is called magnetic induction This links electricity and magnetism in a fundamental way Magnetic induction is also the key to many practical applications

Electromagnetism 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? Experiments were done by Michael Faraday Section 21.1

Faraday’s Experiment Faraday attempted to observe an induced electric field He didn’t use a lightbulb 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 lightbulb 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 Section 21.2

Magnetic Flux, final The magnetic flux can be defined for any surface A complicated surface can be broken into small regions and the definition of flux applied The total flux is the sum of the fluxes through all the individual pieces of the surface The 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 The magnitude of the induced emf equals the rate of change of the magnetic flux The negative sign is Lenz’s Law 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

Applying Faraday’s Law, cont. The emf is produced by changes in the magnetic flux through the circuit A constant flux does not produce an induced voltage The flux can change due to Changes in the magnetic field Changes in the area Changes in the angle The voltmeter will indicate the direction of the induced emf and induced current and electric field Section 21.2

Faraday’s Law, Summary Only changes in the magnetic flux matter Rapid changes in the flux produce larger values of emf than do slow changes This dependency on frequency means the induced emf plays an important role in AC circuits The magnitude of the emf is proportional to the rate of change of the flux If the rate is constant, then the emf is constant In most cases, this isn’t possible and AC currents result The induced emf is present even if there is no current in the path enclosing an area of changing magnetic flux Section 21.2

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 The current leads to power dissipation in the circuit Section 21.2

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 Use rotational motion instead of linear motion A permanent magnet produces a constant magnetic field in the region between its poles Section 21.2

Generator, cont. A wire loop is located in the region of the field The loop has a fixed area, but is mounted on a rotating shaft The angle between the field and the plane of the loop changes as the loop rotates If the shaft rotates with a constant angular velocity, the flux varies sinusoidally with time This basic design could generate about 70 V so it is a practical design Section 21.2

Changing a Magnetic Flux, Summary A change in magnetic flux and therefore an induced current can be produced in four ways If the magnitude of the magnetic field changes with time If the area changes with time If the loop rotates so that the angle changes with time If the loop moves from one region to another and the magnitude of the field is different in the two regions 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 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 downward and the induced current will be counterclockwise Section 21.3

Problem Solving Strategy Recognize the principle The induced emf always opposes changes in flux through the Lenz’s Law loop or path Sketch the problem Show the closed path that runs along the perimeter of a surface crossed by the magnetic field lines Identify Is the magnetic flux increasing or decreasing with time? Section 21.3

Problem Solving Strategy, cont. Solve Treat the perimeter of the surface as a wire loop Suppose there is a current in the loop Determine the direction of the resulting magnetic field Find the current direction for which this induced magnetic field opposes the change in the magnetic flux This current direction gives the sign (direction) of the induced emf Check Consider what your answer means Check that your answer makes sense Section 21.3

Lenz’s Law and Conservation of Energy Mathematically, Lenz’s Law is just the negative sign in Faraday’s Law It is actually a consequence of conservation of energy Therefore, conservation of energy is contained in Faraday’s Law Nowhere in the laws of electricity and magnetism is there any explicit mention of energy or conservation of energy Physicists believe all laws of physics must satisfy the principle of conservation of energy Section 21.3