AP Physics C Chapter 3 Notes Yockers Faraday s aw, Inductance, and Maxwell s Equations Faraday s aw of Induction - induced current a metal wire moved in a uniform magnetic field - the charges (electrons) in the wire will experience a force - the resulting movement of charges produces a current in the wire a stationary wire loop and moving magnet - an electric current is set up as long as relative motion occurs between a magnet and a coil current can exist in a wire even when the wire is not connected to an emf Faraday s experiment - magnetic flux - galvanometer reads zero when there is no current or there is a steady current - the moment the switch is closed or opened a current is detected on the galvanometer the current produced in the secondary circuit occurs only for an instant while the magnetic field acting on the secondary coil builds from its zero value to its final value (when the switch is closed) the changing magnetic field is a result of the current in the primary circuit - an electric current can be produced by a time-varying magnetic field - an emf is induced in the secondary circuit by the changing magnetic field proportional to the number of magnetic field lines passing through an
area the magnetic flux through the element is da - d A is a vector perpendicular to the surface whose magnitude equals the area d A - is the magnetic field at the surface of the area d A - the SI unit of magnetic flux is the weber: 1 Wb = 1 T m - the total magnetic flux through the surface is da emf is induced in a circuit when the magnetic flux through the surface bounded by the circuit changes with time - Faraday s law of induction the emf induced in a circuit is equal to the time rate of change of magnetic flux through the circuit d if the circuit is a coil consisting of N identical and concentric loops and if the field lines pass through all the loops, then d N - the emf is increased by the factor N because all of the loops are in series - the emfs in the loops add to give the total emf if the magnetic field is uniform over the area A bounded by a loop lying in a plane, then da dacos cos da Acos Acos - = the angle between and the direction to the plane of the loop. Why? is at a maximum when plane of the loop is to is at a minimum when the plane of the loop is to enz s aw (the negative sign in Faraday s law) the polarity of the induced emf in a loop is such that it produces a current whose magnetic field opposes the change in magnetic flux through the loop. That is, the induced current is in a direction such that the induced magnetic field attempts to maintain the original flux through the loop - consistent with the continuity equation for energy - a bar moving on two parallel rails in the presence of a uniform magnetic field
- a bar magnet moved with respect to a stationary loop of wire Motional emf arises from the motion of a conductor through a magnetic field - conducting bar of length l moves through a uniform magnetic field due to the magnetic force on the electrons, the ends of the wire become oppositely charged which establishes an electric field in the wire the charge at the ends of the conductor builds up until the magnetic force qv on an electron in the conductor is balanced by the electric force qe on the electron F Fe - F qe qv E v V E v a potential difference is maintained across the conductor as long as there is motion through the field due to charge separation if the motion is reversed, the polarity of the potential difference is also reversed the induced motional emf in the bar is
A x d d dx x v v if the resistance of the circuit is R, the magnitude of the induced current is v I R R - conversion of mechanical energy in the moving bar to internal energy in the resistor power is equal to the rate at which energy is delivered to the resistor power is equal to the power I supplied by the induced emf remember, F F v P F app app v v R - the emf generated by a rotating loop v R R R v R I v R I R as the loop rotates, the magnetic flux through it changes with time, inducing an emf and a current if the loop rotates with a constant angular speed in a uniform - t and v r - f Acos Acos t - the emf is at a maximum when the plane of the of the loop is parallel to the magnetic field d d N NA cos t NA sin t the induced emf is varying and is described by a sinusoidal curve (ac voltage) Induced emfs and Electric Fields - an electric field is created in a conductor as a result of changing magnetic flux - an electric field is always generated by a changing magnetic flux, even in free space where no charges are present
- the induced current in the loop above implies the presence of an induced electric field E that must be tangent to the loop in order to provide an electric force on the charges around the loop the work done by the electric field on the loop in moving a test charge q once around the loop is equal to q because the magnitude of the electric force on the charge is qe, the work done by the electric field can also be expressed as qe r q qer E r using Faraday s law and A r for a circular loop d d r d E 1 1 r r r - this can be used to calculate the induced electric field if the time variation of the magnetic field is known - this result is also valid in the absence of a conductor or charges - the emf for any closed path can be expressed as the line integral of E ds over that path d E ds the induced electric field E in the equation above is a nonconservative field that is generated by a changing magnetic field - it is nonconservative because the work it does moving a charge around a closed path is not zero - this electric field is very different from an electrostatic field Self-inductance - current doesn t immediately jump from zero to its maximum value R as the current increases with time, the magnetic flux through the loop of the circuit
itself due to the current also increases with time the increasing magnetic flux from the circuit induces an emf in the circuit that opposes the change in the net magnetic flux through the loop of the circuit according to enz s law, the induced electric field in the wires must be opposite the direction of the current the opposing emf results in a gradual increase in the current - self-induction the changing magnetic flux through the circuit arises from the circuit itself - the induced emf is, thus, called a self-induced emf - the self-induced emf is always proportional to the time rate of change of the current - for a closely spaced coil of N turns of fixed geometry (a toroidal coil or an ideal solenoid), the proportionality can be expressed as d di N is a proportionality constant called the inductance of the coil - depends on the geometric features of the coil and other physical characteristics - can be determined by N I - can also be written as (usually the defining equation for the inductance of any coil) di - the SI unit of inductance is the henry (H) 1 H = 1 V s/a - resistance is a measure of opposition to current, whereas inductance is a measure of opposition to the change in current R Circuits - a circuit containing a coil has a self-inductance that prevents the current from increasing or decreasing instantaneously a circuit element designed to provide inductance in a circuit is an inductor it is assumed that the self-inductance for the remainder of the circuit is negligible (for this course) a series R circuit as the current increases toward its maximum value, an emf that opposes the increasing current is induced in the inductor - as current begins to increase, after the switch is closed, the inductor produces an emf (sometimes referred to as a back emf ) that opposes the increasing current - the back emf produced by the inductor is di because the current is increasing, di is positive
is negative a potential drop occurs from point a to point b in the diagram above - applying Kirchhoff s loop rule, beginning at the battery di IR the potential across the inductor is given a negative sign because its emf is in the opposite sense to that of the battery - to obtain a mathematical solution to the Kirchhoff s loop rule equation di I R R letting x R I, so dx di dx x R dx R x integrating the last equation from an initial instant to some later time t gives x dx R t x R ln t x x x where the value of x at t is expressed as x R, because I at t x R ln t x x x e x x Rt e Rt which is equivalent to Rt I e R R Rt I 1 e R which is the solution for current from the Kirchhoff s loop rule equation t It 1 e R - where the constant is the time constant of the R circuit - R and has units of time (seconds) V s A V s A s R A V - the steady-state current value of the current which occurs at t is R in steady-state the change in current is zero the final current does not involve because the inductor has no effect on the circuit if the current is not changing the first time derivative of the equation above di t e - the rate of current di is a maximum (equal to ) at t and falls exponentially to zero as t
- an R circuit containing two switches when S 1 is closed and S is open, the battery is in the circuit at the instant S is closed, S 1 is opened and the battery is no longer part of the circuit applying Kirchhoff s loop rule to the S closed S 1 open situation with the current initially at its steady-state value R di IR - the solution of this differential equation is t t It e I e R
the current at t is I R and R the current is continuously decreasing with time the slope of di is always negative and has its maximum value at t the negative slope signifies that di is now positive (point a in the circuit is at a lower potential than point b)
Energy Stored in a Magnetic Field
- applying Kirchhoff s loop rule, beginning at the battery di IR - multiplying each term by the current I gives the rate at which energy is supplied by the battery di I I R I di I I R I I R is the rate at which energy is delivered to the resistor di I is the rate at which energy is delivered to the inductor - if U denotes the energy stored in the inductor at any time, then the rate du at which energy is transferred to the inductor can be written du di I - to find the total energy stored in an inductor at any instant U U 1 U I du I I di - an inductor stores energy in its magnetic field when the current is I Maxwell s Equations - can be regarded as the basis of all electric and magnetic phenomena - represent laws of electricity and magnetism already covered - predict the presence of electromagnetic waves traveling at c - show that electromagnetic waves are radiated by accelerating charges Gauss s law Q E da the total electric flux through any closed surface equals the net charge inside that surface divided by this law describes how charges create electric fields electric field lines originate on positive charges and terminate on negative charges Gauss s law for magnetism d A the net magnetic flux through a closed surface is zero - the number of magnetic field lines entering a closed volume must equal the number leaving - magnetic field lines cannot begin or end at any point isolated magnetic monopoles cannot exist Faraday s law of induction d d E s
the line integral of the electric field around any closed path (which equals the emf) equals the rate of change of magnetic flux through any surface area bounded by that path describes how a changing magnetic field creates an electric field Ampère s law (generalized form) d d I E s the line integral of the magnetic field around any closed path is determined by the net current and the rate of change of electric flux through any surface bounded by that path describes how both an electric current and a changing electric field create a magnetic field - orentz force F q E qv once the electric and magnetic fields are known at some point is space, the force those fields exert on some particle of charge q can be calculated this force and Maxwell s equations describe all classical electromagnetic interactions