Transformers change voltages from one value to another. For example, devices such as cell phones, laptops, video games, power tools and small appliances have a transformer (built into their plug-in unit) that changes 120 V into the proper voltage for the device. Transformers are also used at several points in power distribution systems, as shown in. Power is sent long distances at high voltages, as less current is required for a given amount of power (this means less line loss). Because high voltages pose greater hazards, transformers are employed to produce lower voltage at the user’s location. We have studied Faraday’s law of induction in previous atoms. We learned the relationship between induced electromotive force (EMF) and magnetic flux. In a nutshell, the law states that changing magnetic field(\(\frac { d \Phi _ { \mathrm{B} } } {\mathrm{ d t} }\)) produces an electric field (\(ε\)), Faraday’s law of induction is expressed as \(\varepsilon = - \frac { \partial \Phi _ { \mathrm { B } } } { \partial \mathrm { t } }\), where \(ε\) is induced EMF and \(\frac { d \Phi _ { \mathrm{B} } } {\mathrm{ d t} }\) is magnetic flux. (“N” is dropped from our previous expression. The number of turns of coil is included can be incorporated in the magnetic flux, so the factor is optional. ) Faraday’s law of induction is a basic law of electromagnetism that predicts how a magnetic field will interact with an electric circuit to produce an electromotive force (EMF). In this Atom, we will learn about an alternative mathematical expression of the law. Faraday’s law states that the EMF induced by a change in magnetic flux depends on the change in flux Δ, time Δt, and number of turns of coils. For linear, non-dispersive, materials (such that \(\mathrm{B = μH}\) where μ, called the permeability, is frequency-independent), the energy density is: \(.\mathrm { u } = \frac { \mathbf { B } \cdot \mathbf { B } } { 2 \mu } = \frac { \mu \mathbf { H } \cdot \mathbf { H } } { 2 }\)

The magnetic flux (often denoted Φ or Φ B) through a surface is the component of the magnetic field passing through that surface. mathrm { EMF } _ { 1 } = - \mathrm { M } \dfrac { \Delta \mathrm { I } _ { 2 } } { \Delta \mathrm { t } }\]

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mathrm { EMF } _ { 2 } = - \mathrm { M } \dfrac { \Delta \mathrm { I } _ { 1 } } { \Delta \mathrm { t } }\] A device that exhibits significant self-inductance is called an inductor, and the EMF induced in it by a change in current through it is \(\mathrm{ EMF = −L\frac{ ΔI}{Δt}}\). This is known as the transformer equation, which simply states that the ratio of the secondary to primary voltages in a transformer equals the ratio of the number of loops in their coils. The output voltage of a transformer can be less than, greater than or equal to the input voltage, depending on the ratio of the number of loops in their coils. Some transformers even provide a variable output by allowing connection to be made at different points on the secondary coil. A step-up transformer is one that increases voltage, whereas a step-down transformer decreases voltage.

Energy density is the amount of energy stored in a given system or region of space per unit volume. If there are no magnetic materials around, μ can be replaced by μ 0. The above equation cannot be used for nonlinear materials, though; a more general expression (given below) must be used. Assuming, as we have, that resistance is negligible, the electrical power output of a transformer equals its input. Equating the power input and output, In the most general form, magnetic flux is defined as \(\Phi _ { \mathrm { B } } = \iint _ { \mathrm { A } } \mathbf { B } \cdot \mathrm { d } \mathbf { A }\). It is the integral (sum) of all of the magnetic field passing through infinitesimal area elements dA.

Observations

An electric generator rotates a coil in a magnetic field, inducing an EMF given as a function of time by \(\mathrm{ε=NABw \sin ωt}\). dfrac { \mathrm { I } _ { \mathrm { s } } } { \mathrm { I } _ { \mathrm { p } } } = \dfrac { \mathrm { N } _ { \mathrm { p } } } { \mathrm { N } _ { \mathrm { s } } }\] The EMF produced due to the relative motion of the loop and magnet is given as \(\mathrm{ε_{motion}=vB \times L}\) (Eq. 1), where L is the length of the object moving at speed v relative to the magnet. The energy stored by an inductor is \(\mathrm { E } _ { \mathrm { stored } } = \frac { 1 } { 2 } \mathrm { L } \mathrm { I }

The apparatus used by Faraday to demonstrate that magnetic fields can create currents is illustrated in the following figure. When the switch is closed, a magnetic field is produced in the coil on the top part of the iron ring and transmitted (or guided) to the coil on the bottom part of the ring. The galvanometer is used to detect any current induced in a separate coil on the bottom. mathrm { V } _ { \mathrm { p } } = - \mathrm { N } _ { \mathrm { [ } } \dfrac { \Delta \Phi } { \Delta \mathrm { t } }\]Generators supply almost all of the power for the electric power grids which provide most of the world’s electric power. In this equation, N=1 and the flux Φ=BAcosθ. We have θ=0º and cosθ=1, since B is perpendicular to A. Now \(\mathrm{Δ=Δ(BA)=BΔA}\), since B is uniform. Note that the area swept out by the rod is \(\mathrm{ΔA=ℓx}\). Entering these quantities into the expression for EMF yields: Faraday’s law of induction is a basic law of electromagnetism that predicts how a magnetic field will interact with an electric circuit to produce an electromotive force (EMF). It is the fundamental operating principle of transformers, inductors, and many types of electrical motors, generators, and solenoids. varepsilon = \mathrm { NABw } \sin \omega t\) is the EMF induced in a generator coil of N turns and area A rotating at a constant angular velocity in a uniform magnetic field B.