Electromagnetic Induction AP Physics C

Faraday’s Discovery Electromagnetic Induction is the process of using magnetic fields to produce voltage, and in a complete circuit, a current. Michael Faraday first discovered it in 1831, using some of the works of Hans Christian Oersted. He started by using different combinations of wires and magnetic strengths and currents, but it wasn't until he tried moving the wires that he got any success. It turns out that electromagnetic induction is created by just that - the moving of a conductive substance through a magnetic field.

Magnetic Flux The magnetic flux measures the amount of magnetic field passing through a loop of area A if the loop is tilted at an angle θ from the field. We represent magnetic flux with a dot product, just like with electric flux. The unit for magnetic flux is a Webber. Φ 𝐵 = 𝐵 ∙ 𝐴

Magnetic Flux for Non-Uniform Magnetic Fields If you have a non-uniform magnetic field through a loop, you have to integrate to find the total flux. This often involves a change of variable.

Faraday and Lenz’s Laws Faraday realized that changing the magnetic flux in any way induced a voltage (and therefor a current) in a circuit. Faradays law gives the magnitude of the induced voltage. Lenz’s law (the negative sign) indicates the direction of the induced current. Lenz’s law states that inductors resist changes in magnetic flux. 𝜀=− 𝑑Φ 𝑑𝑡

Lenz’s Law

Example The current in the straight wire is decreasing. Which is true? A. There is a clockwise induced current in the loop. B. There is a counterclockwise induced current in the loop. C. There is no induced current in the loop.

Example The magnetic field is confined to the region inside the dashed lines; it is zero outside. The metal loop is being pulled out of the magnetic field. Which is true? A. There is a clockwise induced current in the loop. B. There is a counterclockwise induced current in the loop. C. There is no induced current in the loop.

Eddy Currents Consider pulling a sheet of metal through a magnetic field. Two “whirlpools” of current begin to circulate in the solid metal, called eddy currents. The magnetic force on the eddy currents is a retarding force. This is a form of magnetic braking.

Motional EMF Motional EMF is voltage generated by a conductive bar moving through a magnetic field. The picture to the right shows a current being generated by motional EMF.

Example An airplane with a wing span of 30.0 m flies parallel to the Earth’s surface at a location where the downward component of the Earth’s magnetic field is 0.60 x10-4 T. Find the difference in potential between the wing tips is the speed of the plane is 250 m/s.

AC Generators AC generators convert mechanical energy into electrical energy. The change in magnetic flux through the loop induces a current, which then goes to powering various devices. This is the basic principle behind wind power and hydroelectric power.

Transformers A transformer sends an alternating emf V1 through the primary coil. This causes an oscillating magnetic flux through the secondary coil and, hence, an induced emf V2. The induced emf of the secondary coil is delivered to the load:

Transformers A step-up transformer, with N2 >> N1, can boost the voltage of a generator up to several hundred thousand volts. Delivering power with smaller currents at higher voltages reduces losses due to the resistance of the wires. High-voltage transmission lines carry electric power to urban areas, where step-down transformers (N2 << N1) lower the voltage to 120 V.

The Induced Electric Field Faraday’s law and Lenz’s law may be combined by noting that the emf must oppose the change in Φm. Mathematically, emf must have the opposite sign of dB/dt. Faraday’s law may be written as:

The Induced Magnetic Field As we know, changing the magnetic field induces a circular electric field. Symmetrically, changing the electric field induces a circular magnetic field. The induced magnetic field was first suggested as a possibility by James Clerk Maxwell in 1855.

Maxwell’s Equations We have now been introduced to the four most important equations in all of electromagnetic field theory. These formulas are known as Maxwell’s equations, named after James Clerk Maxwell.

Electromagnetic Waves A changing electric field creates a magnetic field, which then changes in just the right way to recreate the electric field, which then changes in just the right way to again recreate the magnetic field, and so on. This is an electromagnetic wave (a light wave).

Inductors A coil of wire, or solenoid, can be used in a circuit to store energy in the magnetic field. We define the inductance of a solenoid having N turns, length l and cross-section area A as: The SI unit of inductance is the henry, defined as: 1 henry = 1 H = 1 Wb/A = 1 T m2/A A coil of wire used in a circuit for the purpose of inductance is called an inductor. If you divide both sides by time we get:

Inductors in Circuits When an inductor is placed in a circuit, it may experience either a rise or a drop in voltage. If the current is increasing in the inductor, the voltage decreases. If the current is decreasing in the inductor, the voltage increases. This is a consequence of energy being stored in the inductor’s magnetic field.

Energy in an Inductor As current passes through an inductor, the electric power is: Pelec is negative because the current is losing energy. That energy is being transferred to the inductor, which is storing energy UL at the rate: We can find the total energy stored in an inductor by integrating:

LC Circuits The figure shows a capacitor with initial charge Q0, an inductor, and a switch. The switch has been open for a long time, so there is no current in the circuit. At t  0, the switch is closed. How does the circuit respond? The charge and current oscillate in a way that is analogous to a mass on a spring.

LC Circuits

LC Circuits An LC circuit is an electric oscillator. The letters on the graph correspond to the four steps in the previous slides. The charge on the upper plate is Q  Q0cosωt and the current through the inductor is I  Imaxsinωt, where:

LR Circuits LR circuits involve inductors and resistors. When the switch is closed, the inductor resists changes in magnetic flux, and therefor resists current. As time goes on, the current through the inductor reaches a steady state. The current through the inductor is modeled as logarithmic growth, while the voltage across the inductor is modeled as exponential decay.

LR Circuits If the battery is removed from an LR circuit, the circuit no longer receives emf from the battery, however there is still current for a short period of time. The sudden change in the magnetic flux through the inductor induces an emf whose current can be modeled with an exponentially decaying function. L/R is often represented as τ, which is the time constant for an LR circuit.

Example What is the battery current immediately after the switch has closed? What is the battery current immediately after the switch has been closed for a very long time?