Faraday's Law states that if there is a change of flux through a surface bounded by a loop, there will be an induced EMF in the loop equal to the rate of change of the flux:. If the loop is made of conducting material, there will be an induced current.
The magnetic flux through a surface, as with flux of any field, is the surface integral of the field:. For a uniform field, this is a cross-product of the field and the surface area vector: , where is the area of the surface and is the angle between the field and the area vector (the vector perpendicular to the surface, with the magnitude equal to the area of the surface).
There are three ways that the rate of change of magnetic flux might be non-zero:
The negative sign in Faraday's Law,, indicates that the induced EMF has a sign opposite to that of the rate of change of flux. This leads to Lenz's Law: the induced EMF corresponds to the induced magnetic field that will counteract the change in the flux. That is, if flux increases, the magnetic field of the induced current will be in the opposite direction of the external field. If the flux decreases, the magnetic field of the induced current will be in the same direction as the external field.
Typical problems involving Faraday's Law include:
If the current through a wire loop or a solenoid changes, the magnetic field created by the current also changes. According to Faraday's Law and Lenz's Law, this will cause an electromotive force to be induced in the wire loop or solenoid that is opposite in direction to the electromotive force of the battery supplying the original current. This induced electromotive force will resist the change in current; the tendency of a part of the circuit to provide this resistance is called inductance.
Inductance is the proportionality coefficient between the induced EMF and the rate of change of current:. This is because the induced EMF is equal to the rate of change of magnetic flux, and the flux is proportional to the current. The inductance depends on the geometry and magnetic properties of the part of the circuit but is independent of the current. For example, the inductance of a solenoid with a number of turns , length , and cross-sectional area is . The part of the circuit that provides resistance to a change in current is called an inductor. Inductance is sometimes called an "electrical inertia", since it measures resistance to the change in current similar to the way mass, or mechanical inertia, measures resistance to change in velocity.
The property of inductance as related to a magnetic field is analogous to capacitance. Just like how a capacitor can contain an electric field and store electric field energy, an inductor can contain a magnetic field and store magnetic field energy. The magnetic energy stored inside an inductor with inductanceand current is . Notice that this expression is similar in form to the one for kinetic energy ( ) and the one for electric energy stored inside a capacitor ( ).
In an RC circuit, as a battery is connected to an uncharged capacitor through a resistor, the charge on the capacitor will build up exponentially until it reaches its maximum value The RC Circuit., where is the voltage supplied by the battery: Q=Qmax(1 - e-t/). Here, the constant =RCis called the time constant of the RC circuit. In indicates the time when the charge on the capacitor reaches about two-thirds of its final, or maximum, value. If the battery is shorted out of the circuit, the capacitor will discharge through the resistor and its charge will exponentially approach zero: . To review how to analyze RC circuits, refer to
In an RL circuit, as a battery is connected, the current will tend to increase, but its growth will be slowed down by the inductor. The current will rise exponentially until it reaches its maximum, constant value of The RL Circuit., where is the voltage supplied by the battery: . Here, the constant is called the time constant of the RL circuit. It indicates the time when the current in the circuit rises to about two-thirds of its final value. If the battery is shorted out of the circuit, the current will exponentially decay to zero: . To review how to analyze RL circuits, refer to
In an ideal LC circuit (consisting of an inductor and a capacitor with negligible resistance), the charge on the capacitor will oscillate, as will the current in the circuit. This current is the electronic equivalent of a mechanical system that undergoes simple harmonic motion, such as a mass on a spring that undergoes. If atthe capacitor is fully charged to charge , the charge will depend on time with a cosine function: . The angular frequency of the oscillations (the natural frequency of the system) is . The energy in this circuit is conserved, as it is periodically converted from electric energy by the capacitor to magnetic energy by the inductor.
Since all real circuit elements have resistance, circuits used in real-world applications are RLC circuits. Here, energy dissipates on the resistor, and the amplitudes of charge and current gradually decrease. If, however, a circuit is connected to an AC voltage source, that source will supply the energy to keep the oscillations going. The maximum amplitude of the current in the circuit is achieved when the frequency of the AC voltage source (the driving frequency) equals the natural frequency of the system, Light and Matter.. This phenomenon is known as resonance. Read about RLC circuits and their analogy to mechanical systems in sections 25.2 and 25.3 of
Electromagnetic generators are wire loops rotating in an external magnetic field that is usually created using bar magnets. As the angle between the surface of the loop and the magnetic field lines changes, the magnetic flux through the loop changes, which induces an electromotive force and current in the loop. The rotational kinetic energy of the loop is thus converted to electrical energy and generates an AC current of the same frequency as the frequency of the rotation of the loop. To produce DC current, a commutator is required, which changes the direction of the EMF every half-period. The basic principles of the operation of generators are described in The Alternating Current Generator and The Direct Current Generator.
In a motor, the torque of the external magnetic field, usually provided by bar magnets, rotates the current-carrying loop of wire, thus converting electric energy into mechanical (kinetic rotational energy). If the current is constant, the torque will simply flip the loop only once, so there has to be a way to change the direction of the current: supply alternating current to the loop, or connect the loop to a commutator which will change the direction of the current periodically. Then, the continuous rotation will be established. However, there will be electromotive force induced in the loop rotating in the magnetic field, which will tend to slow down the rotation. Additional modifications are required in order to counteract that effect. The basic principles of operation of motors are described in Professor Fitzpatrick's articles The Alternating Current Motor and The Direct Current Motor. Also, watch Electric Motors.
This vocabulary list includes terms that might help you with the review items above and some terms you should be familiar with to be successful in completing the final exam for the course.
Try to think of the reason why each term is included.