Electricity and Magnetism Flashcards
how did Faraday discover EM induction
a voltage was produced when magnetic field through circuit changed
modern version of Faraday’s experiment
- magnet at rest = no current in coil
- magnet moving relative to coil = current induced in coil
- second current-carrying coil moving relative to stationary coil = current induced in stationary coil
- second current-carrying coil at rest relative to outer coil = current induced in outer coil only when current in inner coil changes
magnetic flux
similar to electric flux
divide any surface into elements of area dA
magnetic flux through surface element is B.dA
total magnetic flux is sum of these contibutions i.e integral of B.dA
units of magnetic flux
Tm^2 = webber Wb
Faraday’s law
induced emf in a coil is proportional to the negative of the rate of change of magnetic flux through the coil
for coil with N turns, emf is N times bigger
emf
voltage generated by battery or by magnetic force
energy per unit charge
available by chemical energy for battery, magnetic for induction
B.A=BAcostheta where theta is
the angle between the direction of the area vector (normal to the plane of the coil) and the direction of the magnetic field
what can give rise to an induced emf
changing any of B, A or theta
right hand rule
- point thumb of right hand in +ve direction of area vector
- induced emf is +ve in direction of curled fingers
- positive charges in direction of emf
if emf is induced in a closed loop
it will cause a current
direction of induced conventional current
in same direction as induced emf
Lenz’s law
electrons flowing along wire represented as a current flow (+ to -) in opposite direction
current flow creates a magnetic field around wire
how to work out field direction
RH grip rule
RH grip rule
grip wire with right hand
point thumb in direction of (conventional) current flow
fingers wrap in direction of magnetic field
current induced by
a change in magnetic flux
current induced by a change in magnetic flux creates additional…
magnetic field, B induced
the induced current produces a magnetic field that tends to…
oppose the change in magnetic flux that gave rise to the induction
eddy current created when
a moving conductor experiences changes in a magnetic field generated by a stationary object
or when a stationary conductor encounters a varying magnetic field
relative motion causes circulating eddies of current within the conductor which in turn…
create magnetic fields that oppose the effect of the applied magnetic field
eddy currents create losses due to
resistive heating (ohmic heating) as the heating power generated in an electrical conductor of resistance E through which a current I is flowing if P prop to I^2R
eddy current use
brakes on trains
how to reduce eddy currents
thin laminations (insulated layers) that minimise current flow
induced emf - area of coil is A and magnetic field strength is B so magnetic flux through coil is
ΦB = B.A = BAcos theta
induced emf - as the loop is rotating uniformly Φ(t)=
Φ0 + wt
induced emf - according to faraday’s law ε=
- dΦB/dt = BAwsinΦ
this is the basis of AC generators
AC generator
mechanical energy input to a generator turns the coil in the magnetic field
voltage proportional to the rate of change of the area facing the mag field is generate in the coil
sinusoidal voltage output
commercial AC generators
single power station sized generator
B field provided by electromagnet to get large B
reverse stationary and rotating bit so rotating B field and stationary coils
slidewire generator - flux through the loop is changing because
area of the loop in the uniform field is changing as the rod is slid to the right
induced emf in slidewire generator
A and B point into page so ΦB=BA
area and width increasing uniformly
A(t) = Lw(t) = A0 + Lvt
faraday’s law: ε= - dΦB/dt = -d/dt[B(A0+LVT)] = -BLv
magnitude of induced emf in slidewire generator
-BvL
the -BvL emf was caused only by
moving conductor
isolated conducting rod emf
same emf developed between its ends
force on a charge Q in magnetic field B
F=Qv x B
emf between ends of rod
ε= integral of E.dL where path integral is along the rod
motional emf - using F=QE
sub in E=F/Q to integral of E.dL
and then sub in F= Qv x B and then cancel Qs
so ε=vBL
changing flux causes an induced emf and hence current, how?
changing magnetic flux causes an induced electric field in the conductor, must be non-conservative since it does net work on a charge as it is driven around loop
i.e. integral E.dL = - dΦB/dt
changing current will give rise to
changing magnetic field produced by the first circuit and hence a changing magnetic flux through the second
Faraday’s law then tells us there will be an induced current in the second circuit
induction between two adjacent coils
current 1 in coil 1 creates a magnetic field in region of coil 2
magnetic flux through coil 2 is N2ΦB2 (N is no of turns in coil) which is prop. to B and therefore I1
mutual inductance
proportionality constant M such that
N2ΦB2=MI1
unit Henry (H)
what does value of mutual inductance depend on
properties (geometric etc) of the two coils
reciprocity theorem
other way round same thing ie
N1ΦB1=MI2
solenoid
used interchangeably with coil
induction happens when
current in one of the coupled coils changes
mutual inductance and emf
start with N2ΦB2=MI1
take time derivative of both sides
use faraday’s law to get emf induced in coil 2
(same applies if started with other way round)
transformers - if current flowing through coil 1 is I1=I0coswt, emf induced in coil 2 is
ε2= -MdI1/dt = MI0wsinwt
induction applications
transformers used: powering low-voltage devices from mains, generation of high voltage eg ignition sparks in car engines
also used: metal detectors, wireless chargers and security tags
self inductance L
Proportionality factor depends on number of turns, geometry and material inside coil, unit is henry
NΦB=LI
flux through a single turn
ΦB=BA
=μ0NI/l piR^2
if inductor has a material in its core, we need to
replace vacuum magnetic permeability μ0 with μrμ0
if the current in the inductor varies, then so does
the flux through its N turns
NΦB=LI
(take time derivs and use faraday’s law)
potential change or voltage across an inductor
ε=-N dΦB/dt = -L dI/dt
electric field lines
originate at positive charges and direction is direction of the net field at that point
never cross each other
density of field lines at a location indicates magnitude of field there
outward electric flux
positive charge inside the box
electric flux
analogous to magnetic flux
ΦE= integral E.dA
Gauss’ law
electric flux over the surface of a volume V is prop. to electric charge contained in V
ΦE= integral over surface of V E.dA = Qenclosed/ε0
an inductor opposes…
(by generating the induced emf) any change in the
current flowing through it
So work must be done by an external source to change the current
resistors vs inductors
resistor with current I: energy is dissipated
inductor with current I: energy is stored
- Going round a loop, if an inductor is traversed in the same direction as
the direction of flow of positive (conventional) current
the “potential
change” (the potential at the output node of the inductor minus the
potential at the input node) is
-L dI/dt
- Gauss’s law provides an easy way of finding
the electric field
for charge distributions that exhibit a high degree of
symmetry
symmetry in some problems allows an intelligent choice of
integration (Gaussian) surface
General recipe for using Gauss’s law to calculate E
1.From the symmetries of the problem, deduce the direction of the
electric field you want to calculate
- Set up a surface (“Gaussian surface”), chosen such that the
calculation of the electric flux through the surface is easy - Compute the electric flux through the Gaussian surface, which will
involve the unknown electric field E as a variable - Equate the flux to (enclosed charge)/ε0
- Calculate E
in equilibrium, the electric field inside a conductor is
zero, and any excess
charge on the conductor resides solely on its surface
the work done by the E field when moving a charge from A to B is
path independent
the component of the E field that is parallel
to the surface is
continuous, the general boundary condition
on the E field
electrostatic force is a
conservative force
work done by the force round a closed path is
0
potential for a conservative force
potential energy per unit charge, volts
work done by the electric force when a unit charge moves
from a to b
Vab= Va - Vb
equipotentials
always normal to electric field lines
(No work to move charges
along equipotentials so no
electric field component
along equipotentials)
capacitor
A capacitor is a device for storing electric charge
* It has two conductors which are electrically separated
* One is positively charged, the other negatively
various geometric possibilities for a capacitor including:
- Parallel plates
- Coaxial cylinders
- Concentric spheres
- If charges +Q and –Q are present on the two plates there will be a
potential difference V between the plates
* The plate with +ve charge is at higher potential
The electric field between the plates of a capacitor is proportional to
the magnitude of Q
so potential difference between the plates is also proportional to Q
capacitance is defined by the ratio
C = Q/V
SI unit of capacitance
Farad
1F=1C/V
1 farad is a huge capacitance
the potential difference between the plates is given by
V=Ed
instead of vacuum, most practical capacitors have
an insulating material between the electrodes
such material is called a dielectric
why use a dielectric
- makes it easy to get conductors close together (larger capacitance) without danger of electrical contact
- gives bigger capacitance because of what dielectrics do to the electric field inside the capacitor
- can withstand higher voltages than air allowing higher stored charge and energy
what does a dielectric do?
adding a dielectric to a charged capacitor reduced the potential difference between plates
removing it restores potential difference
if charge stays the same with a dielectric and voltage reduces then…
capacitance must be increased
without dielectric: C0=Q/V0
with: C=Q/V
dielectric constant (or relative permittivity) of material
εr=C/C0
for fixed charge, presence of a dielectric reduces the
potential difference by a factor of εr
V=V0/εr
for fixed free charge on the plates, if the voltage across the plates decreases then so
must the electric field in the gap
E=E0/εr
net surface charges
all +ve charges slightly shifted one way, -ve other way
surface charges not free to move ie bound charges
**dielectric is polarised **
if a dielectric with dielectric constant εr is inserted between the plates, electric field becomes
E=E0/εr= Q/A / εrε0
ε=εrε0 is permittivity of the material
capacitance for parallel capacitors
voltage for both capacitors is same so
Ceq=Q/V = Q1+Q2/V = Q1/V +Q2/V = C1 + C2
capacitance for capacitors in series
Ceq= Q/V = Q/V1+V2
1/Ceq = V1+V2/Q = V1?Q +V2/Q = 1/C1 +1/C2
combining capacitors
think of like combining resistors with rules:
parallel - Ceq=C1+C2
series - Ceq= 1/C1 + 1/C2
rate of charge transfer to capacitor’s left plate
I=dQ/dt
RC circuit charging - voltages (loop rule)
ε = VR+VC
sub in VR=IR
VC=Q/C
I=dQ/dt
end up with diff.eqn.
energy stored in capacitor is energy stored in
the electric field
In a plate capacitor, this E-field fills a volume Ad resulting in an enery density
energy density in magnetic field
uB = 1/2B^2/μ
μ is permeability
energy density in electric field
uE=1/2 εE^2
ε is permittivity
in AC circuits, voltage and current vary
sinusoidally
AC circuit sinusoidal form
V(t)=V0cos(wt+phi0)
AC allows more efficient
power transmission (because of transformers)
phasors
graphical representations of AC quantities
vector rotating anticlockwise
instantaneous value is projection onto horizontal axis
phasor - resistor across AC source
V0/R cos(wt+phi0)
voltage and current in phase
parallel on phasor diagram
phasor - inductor across an AC source
I(t)=I0cos(wt+phi0)
V(t)=L dI/dt = -LI0w sin(wt+phi0)
voltage leads current by 90 degrees
phasor - capacitor across an AC source
V(t)=Vocos(wt+phi0)
I(t)=dQ/dt = d/dt CV(t) = -CV0wsin(wt+phi0)
voltage lags current by 90 degrees
Lenz’s law tells us that induced emf resists…
the buildup of current when a voltage is applied to an inductor
takes a finite time for current to rise to max
so voltage across inductor leads current
for a capacitor, the current that flows to charge up capacitor, causes voltage
across the capacitor to rise
so voltage lags the current
rms voltage
Vo/root 2
rms values are important for calculating
dissipated power when AC signals are used
impedance, Z
ratio of complex voltage and current
Z=V/I
impedance for a resistor
Z=R
simply resistance and it is real
impedance for inductors and capacitors
Z is complex describing both the relative magnitude and phase of voltage and current
impedance of an inductor
start with sinusoidal current
I=I0exp(iwt)
voltage is V=L DI/dt = iwLI
Z=V/I = iwL
impedance of a capacitor
start with sinusoidal voltage
V=V0exp(iwt)
Q(t)=CV(t)
I=dQ/dt = iwCV
Z=V/I = 1/iwC
Z for resistor
Z=R
Z for inductor
Z=iwL
Z for a capacitor
Z= 1/iwC
phasor - impedance for resistor
V and I parallel
phasor - impedance for inductance
V leads I by 90 degrees
phasor - impedance for capacitor
V lags I by 90 degrees
impedances in series
exactly like resistors
impedance in parallel
exactly like resistors
series LCR circuit
loop rule
differentiate wrt t
divide by L
use I(t)=dQ(t)/dt
series LCR circuit is same as
damp, driven harmonic oscillator
LC circuit
energy oscillates between being
stored in capacitor (in E field) and inductor (B field)
impedance analysis
impedance of all the components adds up (series)
Z=R+L+C
greatest current
resasonace
I=V/Z
maximum |I| when
|Z| is minimal
what w when |Z|minimal?
w=w0=1/sqrt(LC)
expensive way to have voltage source whose voltage is independent of current
build huge redundancy into voltage source
alternative way to have voltage source whose voltage is independent of current
use feedback to keep voltage constant
standard feedback component: op-amp
key parts of an op-amp
non-inverting input V+
inverting input V-
voltage source,G=open-loop gain
Vout
output of opamp is
input multiplied by a factor of G
ideal open loop gain
infinity
ideal input resistance
infinity
ideal output resistance
0
negative feedback example
if bicycle is heading too far right, you steer left
non-inverting amplifier
voltage source attached to non-inverting part
infinite input impedance means
no current flowing through inputs
infinite G means
voltage at inverting and non-inverting inputs are the same
exact parameters of op-amp don’t matter as long as
G»1
Rin»1
Rout«1
wide tolerances of op-amp mean
very cheap to manufacture
