Faraday's Law, Lenz's Law and Inductors

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This is a topic from Higher Physics 1B

Contents

Introducing Faraday's Law

Introducing Magnetic Induction and Electromotive Force

  • It is given that a moving charge or current induces a magnetic field, however Michael Faraday showed that a varying magnetic field induces a current
  • Magnetic induction occurs when there is a relative motion between a conductive wire and the source of a magnetic field - effectively induction is caused by a change in magnetic flux
  • The induced current necessitates the existence of an induced voltage known as Electromotive Force (EMF) which is not a force as the name might suggest, but refers to potential energy per unit charge (the same definition as voltage)
  • Faraday's Law describes the induced EMF associated with magnetic induction

Statement and Equation

Faraday's Law of Induction states that the induced electromotive force (EMF) in any closed circuit is equal to the negative of the time rate of change of the magnetic flux through the circuit:

Screen Shot 2012-08-20 at 12.44.25 PM.png

Where ε represents EMF in Volts (V)

ΦB represents magnetic flux in webers (Wb) or Tesla-metres-squared (Tm2)
t represents time in seconds (s)


Substituting in the definition of magnetic flux, and accounting for the number of loops of wire (N) gives:

ε = -N d(BA Cosθ) / dt

and so upon analysis we see that an EMF can be induced by a change in the area, the angle, the magnetic field strength and even the number of loops.

Motional EMF

A motional EMF is one induced in a conductor moving through a constant external magnetic field. One way of conceptualising the induced current is to consider that given that the electrons within the conductor are moving in an external magnetic field they undergo a force (F = qvB) the direction of which is the direction of the induced current. This results in a potential difference between the two ends of the conductor, and an electric field between them. Thus when there is equilibrium:

F = qE + qvB = 0
qE = qvB
E = vB

The electric field generated over the length of the conductor ɭ is related to potential difference by:

ΔV = Eɭ
ΔV = Bɭv

Magnetic Force on a Sliding Bar

Consider the following setup:
Screen Shot 2012-08-20 at 2.00.22 PM.png
Considering force:

  • There is a change in flux because the flux is given by ΦB = BA = Bɭx and x varies with time
  • It can be rewritten that:
ε = -dΦB / dt
= -d(Bɭx) / dt
= -Bɭ (dx / dt)
= -Bɭv
  • Because V = IR:
I = |ε| / R
= Bɭv / R
  • FB = -IɭB
  • Applying Newton's 2nd Law:
Fx = ma = -IɭB
m(dv / dt) = -(Bɭv / R) (ɭB)
dv / v = -(B2ɭ2 / mR) dt
dv / v = -(B2ɭ2 / mR) dt
  • Integrating both sides using the initial condition that v = vi at t = 0 and noting that B2ɭ2 / mR is constant:
ln(v / vi) = -(B2ɭ2 / mR) t
  • Therefore:
v = vie-t/τ where τ = mR / B2ɭ2


Considering power:

  • Power is the rate at which energy (work) is delivered to the resistor:
P = ΔE / Δt = ΔW / Δt = F dx / dt = Fv
  • Substituting F = IɭB:
P = IɭBv
  • Substituting I = Bɭv / R:
P = B2ɭ2v2 / R = ε2 / R

Lenz's Law

Lenz's law describes the direction of the current induced by Faraday's Law. It states that the induced current in a loop is in the direction such that the resultant magnetic field opposes the change in flux which produced it (the current attempts to keep the original magnetic flux by opposing changes to it).

Example:
Fig198.gif

  • In the above setup the ring begins to fall due to gravity, causing a relative velocity and a change in flux
  • A current is induced in the ring by Faraday's law
  • The direction of the current is such that the resultant magnetic field resists the velocity/change in flux
  • This means the resultant magnetic field is directed downwards
  • This requires the current to be flowing clockwise when viewing the ring from above

Induced EMF and Electric Fields

The existence of an induced EMF necessitates the existence of an electric field, however this electric field (unlike the electric field resulting from stationary charges) is non-conservative, with the work done in moving a charge around a conducting loop to its initial position is qε. The work done can also be given by:

W = Fx = qEx = qE(2πr)

And so equating these two expressions for work gives:

E = ε / 2πr

Using this result and the magnetic flux through a circular area ΦB = BA = Bπr2 gives:

E = ε / 2πr = (-1 / 2πr) (dΦB / dt)
E = (-r / 2) (dB / dt)

It can also be shown that:

W = Fx = qExperimeter
qExperimeter = qε
Experimeter = ε
E ds= ε
E ds= - dΦB / dt - This is Faraday's Law in general form

Motors and Generators

Generators

Generators convert work into electrical current. Consider the following AC generator:
Screen Shot 2012-08-20 at 3.30.58 PM.png

  • As the loop rotates in the magnetic field the angle it makes with the field changes and so the perpendicular area changes
  • This means there is a change in flux
  • It follows that a current is induced which resists the rotation by Lenz's law
  • By Faraday's Law:
ε = -N d(BA Cosθ) / dt
ε = -NAB d(Coswt) / dt (w is the angular speed)
ε = NABw Sinwt
  • This relationship between EMF and time explains the sinusoidal graphs of current and EMF against time:
Screen Shot 2012-08-20 at 3.37.30 PM.png
  • It follows that εmax = NABw
  • For a DC generator the two ring contacts are replaced with one split-ring commutator which reverses the electrical contact every half rotation such that the output is all in one direction:
Screen Shot 2012-08-20 at 3.41.47 PM.png

Motors

Motors are devices which convert electrical energy into kinetic energy (work). It is essentially a generator running in reverse whereby the current is supplied to the loop which experiences a torque due to the external magnetic field and so rotates consistently in one direction.

Eddy Currents

Eddy currents refers to the circular currents induced in a bulk piece of metal which arise due to Lenz's law. The currents produce a magnetic field which resists the change in flux which created them. This results in an often undesireable conversion of mechanical energy into internal energy. An example of Eddy currents is the following pendulum with a conducting sheet in which Eddy currents are induced such that the pendulum slows down every time it passes through the magnetic field:
Screen Shot 2012-08-20 at 3.49.45 PM.png Eddy currents can be reduced by cutting slots into the conductor which minimise the size of Eddy current loops.

Inductors and Self-Induction

Self-Induction

  • When the switch is closed in a circuit, current begins to flow
  • This current causes a magnetic field to flow through the area of the circuit
  • This causes an increase in flux which induces an EMF by Faraday's law, which opposes the change by Lenz's law, such that the EMF is in the opposite direction to the EMF of the battery which caused the flow of current
  • This self-induction is known as back EMF and it results in a gradual (not immediate) increase in current from 0 to the maximum of |ε| / R
  • During this process work is being done on the wire which is acting as the inductor, such that energy is stored in the form of a magnetic field
  • The self-induced EMF is given as follows:
εL = - L (dI / dt) where L is the constant of inductance whereby
L = NΦB / I = - εL / (dI / dt)

Units of Inductance

The unit of inductance is the henry (H) whereby 1 H = 1 VsA-1

Inductors

  • An inductor is a component added to a circuit for its high coefficient of inductance
  • It increases the initial resistance to the flow of current when the switch is closed, but the result is that energy is stored in the form of the induced magnetic field
  • If the switch is opened again the magnetic field will momentarily cause current to continue to flow by exerting a force on electrons in the conductor. Essentially it is releasing its stored energy during this process
  • Inductors, like capacitors, store energy, however while capacitors store energy through their electric fields, inductors store energy through their magnetic fields

Inductance in a Solenoid

The inductance for a solenoid is derived as follows:

  • By Faraday's Law, for several coils, the induced E.M.F due to a change in flux is given by:
ε = -N(ΔΦB / Δt) = -NA(ΔB / Δt)
B = μoNI / ɭ
[ɭ here represents length, while I represents current]
  • Therefore substituting this into the first equation:
ε = (-μoN2A / ɭ ) (ΔI / Δt)
  • Also by the definition of inductance:
ε = -L (ΔI / Δt)
  • Thus the inductance of a solenoid is given by:
L = μoN2A / ɭ

Energy in a Magnetic Field

  • In a circuit the voltage supplied by the battery can be considered to be an EMF such that the equation for power (P = IV) becomes P = Iε.
  • Therefore in a circuit containing both a resistor and an inductor, power supplied by the battery is dissipated across the resistor R, and stored in the inductor L:
P = εI = I2R + εLI

but εL = L (dI / dt) and so:

εI = I2R + LI (dI / dt)
  • If U represents the potential energy stored by the inductor:
P = dE / dt = dU / dt = LI (dI / dt)
dU = LI dI
  • Integrating gives:
U = LI2 / 2

Energy Density of a Magnetic Field in a Solenoid

  • Given that U = LI2 / 2, substituting L = μoN2A / ɭ and B = μoNI /ɭ gives:
U = (B2 / 2μo)(Aɭ )
  • And since Aɭ is the volume of the solenoid, the magnetic energy density μB is given by:
μB = B2 / 2μo
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