Electromagnetism Flashcards
Equation for current.
I = dq/dt
Coulomb’s law.
F = k Q1Q2/(r^2)
How does Coulomb’s law change for vector?
Multiply by vector r^
How can the force on a charge be found from two close point charges?
Using superposition principle.
Equation for electric field.
E=F/q
E and F are vectors.
How can electric fields from point charges be combined?
Using superposition principle.
Dipole moment equation
p=qa
p is vector
a is vector from negative to positive
Derive equations for electric field parallel and perpendicular to the dipole axis.
Check derivation.
Derive equation for electric field of a dipole in a general position.
Check derivation.
By superposition, derive the field from an infinite plane of uniform charge distribution.
Check derivation.
Flux equation
Flux = ∫E.dS
Gauss’s Theorem
Sum(charge)/ε0 = ∫E.dS
Easy geometries for Gauss’s Theorem
Field perpendicular to surface and same value at all positions on surface.
Field parallel to surface (E.dS=0)
Work done moving charge in electric field
W=-q ∫E.dl
As dW=F.dl
and F=-qE when external force is applied
Work done moving charge through closed loop
=0
Due to conservative nature of electric field
Derive an expression for potential energy of a dipole in an E-field.
Check derivation
How can the potential from two point charges be found?
Using superposition.
How to find E from V?
Partially differentiate V to give E in a certain direction.
E=-gradV
How are equipotentials distributed in a conductor?
There is no potential difference in a conductor thus a conductor carrying static charge is an equipotential volume with an equipotential surface.
Definition of capacitance.
Ratio of charge to potential
q=CV
Energy stored in a capacitor equation.
dW=Vdq
W = ∫Vdq = QV/2
Equation for εr
εr=Cd/C0
Capacitence with dielectric to capacitance without.
Define electric polarisation
Dipole moment per unit volume
Total E field in presence of dielectric
E=E0+Ep
In isotropic materials, equation relating polaristation and field.
P=ε0χE
χ is electric susceptibility.
Derive relationship between dielectric constant and susceptibility.
Check derivation
Result
εr=1+χ
Gauss’s Law in presence of a dielectric
∫E.dS=sum qc/ε0 + sum qp/ε0
Derive D = ε0E+P
Check derivation
Derive D = ε0εrE
Check derivation
Derive boundary conditions for an electric field.
Check derivation
Derive boundary conditions for electric displacement.
Check derivation
What are piezoelectrics?
Materials that exhibit electric polarisation under the influence of applied mechanical stress.
What are pyroelectrics?
Materials in which the electric polarisation can be modified by changing temperature.
What are ferroelectrics?
Non-linear dielectric materials with spontaneous polarisation that can be controlled and reversed by the application of an electric field.
What is a necessary condition of piezoelectrics, pyroelectrics and ferroelectrics?
The crystal structures do not contain a centre of symmetry.
Piezoelectric effect expressed in two equations.
P = qZ+χε0E
P is polaristaion, q is piezoelectric coefficient, Z is stress, χ is electric susceptibility, E is electric field.
e = sZ+qE
e is strain and S is compliance
Why must a material be a single crystal for it to display piezoelectric effects?
In a polycrystalline material the dipoles will cancel.
Equation for pyroelec tric coefficient.
p=dPs/dT
p is pyroelectric coefficient, Ps is spontaneous polarisation and T is temperature.
Applications of pyroelectrics.
Temperature sensors (though unreliable due to air) Detectors and infra red imaging devices.
Derive the condition for the ferroelectric state to be achieved from the ferroelectric catastrophe.
Check derivation
Area inside a ferroelectric hysteresis loop.
Is equal to the energy disipated.
Current in terms of current density equation.
I=∫J.dS
Define current density.
Charge oer unit time crossing a unit area of surface which is perpendicular to the direction of motion.
Derive I=-neAv by differentiation and hence derive J.
Check derivation.
Show that resistance is proportional to length.
Check derivation.
Show that resistance is inversely proportional to area.
Check derivation.
Microscopic form of Ohm’s law
J=σE
E=ρJ
Derive R=ρl/A for a one dimensional current flow through wire length l and CSA=A
Check derivation
Derive power disipated for a small charge dq being taken through a potential V in time dt.
Check derivation
State the assumptions of the Drude model.
Electrons are a ‘gas’
Electrons are only scattered by collisions with ionic cores
Electrons do not interact with each other or ions
The average time between collisions is τ with probability of scattering per unit time of 1/τ.
Derive σ using the Drude model
Check derivation
Strengths of the Drude model.
Accurately opredicts average scattering distance.
Weaknesses of the Drude model.
Underestimates electron velocities and can’t accurately predict thermal conductivities or Cp.
Does not use quantum mechanical model.
Two main consequences for practical applications of superconductors.
Currents generated in superconducting wires do not need a power supply to keep them going.
Currents in superconductors do not generate hear, meaning higher current densities can be maintained compared to normal wires.
State Kirchoff’s first law.
The algebraic sum of the currents meeting at a junction equals zero.
State Kirchoff’s second law
The algebraic sum of potential differences around a closed path in the circuit equals the EMF along the path.
Derive the equation for an unknown resistance of resistor in a Wheatstone bridge.
Check derivation
Derive F=IdlxB
Check derivation
Derive Lorentz force on charge q.
Check derivation
Equation for magnetic dipole moment of a bar magnet
m=qA
q is pole strength.
Equation for magnetic dipole moment if a current loop
dm=IdA
m=IA
By considering force on a rectangular current loop, derive an expression for the torque acting on the loop.
Check derivation
State the Biot-Savart Law
Check
Use Biot-Savart Law to derive magnetic field around a straight wire carrying steady current.
Check derivation
Use Biot-Savart Law to derive magnetic field along axiz of a circular loop.
Check derivation
Use Biot-Savart Law to derive magnetic field along the axis of a solenoid
Check derivation
Equation for force between two monopoles.
F = (μ0q^2)/(4πr^2)
Equation for magnetic field around a monopole.
B = (μ0q)/(4πr^2)
Work doen in moving a monopole in a magnetic field B a distance dl.
dW=B.dl
Is a magnetic field conservative?
No, meaning W=∫B.dl≠0
State Ampere’s law
The work done in taking a unit monopole around a closed path is equal to μ0 times the current threading the path.
Ampere’s law equation
∫B.dl = μ0I
Use Ampere’s law to find the magnetic field due to a solenoid.
Check derivation
Use Ampere’s law to find the magnetic field due to a toroidal coil.
Check derivation
Definition of magnetic flux
Amount of magnetic field crossing a given surface.
Equation for flux
Φ =B.A
Gauss’s theorem for a B field.
∫B.dS = 0
The flux of B over a closed surface is zero.
Why is the flux of B over a closed surface zero?
A magnetic monopole cannot be isolated.
State Faraday’s Law
Induced emf depends on rate of change of flux
Lenz’s law equation
EMF = dΦ/dt
State Lenz’s law
Wherever a change in flux induces a current, the direction of the current flow is such as to oppose the change that caused it.
Derive Faraday’s law from force on a moving charge moving at velocity v in a field strength B.
Check derivation
Equation for self inductance
Φ=LI
Definition of self inductance
The magnetic flux passing through a circuit when a unit current flows in it.
Derive the self inductance in a long solenoid.
Check derivation
What is mutual inductance?
Where two circuits are close to one another and flux from one influences the other and vice versa.
Find the mutual inductance in two coils of same radius with different numbers of turns.
Check derivation
Equation linking mutual inductance and self inductance in transformer.
M^2=L1*L2
Not gernally applicable, if circuits moved apart then M^2
Equation for relative permeability of an inductor with a magnetic material present Lm.
Lm/L0=μr
Relate B and H using μr and μ0
B=μrμ0H
Relate B to H and M
B=μ0(H+M)
Where M is magnetic moment per unit volume (megnetisation).
Relate M and H.
M=χmH
χm is magnetic susceptibility
What are the boundary conditions for magnetostatics?
B1 perp = B2 perp
H1 para = H2 para
Potential energy for a magnetic dipole in a B field.
U = -m.B
negative sign due to dipole lowering energy as it aligns with field direction.
What is a diamagnetic material?
A material with no permanent dipole moments on the atoms.
What happens when a diamagnetic material is placed in a magnetic field?
EM induction takes place and the electrons orbit nucleus to generate an opposing magnetic field. This effect is weak so diamegnetic materials have small χ values.
For a diamagnetic material’s χ value vary with temperature?
Not significantly.
A superconductor has a χ value equal to what?
χ=1
as perfect diamagnet.
What is a paramagenetic material?
A material in which permanent dipole moments exist from electron spin but the dipoles are randomly arranegd so in the absence of a magnetic field there is no magnetisation.
What happens when a magnetic field is applied to a paramagentic material?
The dipoles tend to align in the same direction as the applied field to lower their energy.
What kinds of χ values do paramagnetic materials have?
Small positive χ values.
How does χ vary with temperature for a paramagnetic material?
χ = C/T
C is a positive constant
What is a ferromagnetic material?
A material where below a critical temperature, neighbouring atoms interact (exchange interaction) causing magnetic moments to align parallel to one another. This is spontaneous magnetisation (magnetisation without an external applied field).
What is a Curie temperature for a ferromagnet?
The temperature below which the material displays ferromagnetic properties.
How does a ferromagnetic material behave above the Curie temperature?
Like a paramagnetic material.
How does χ vary with T for a ferromagnetic material?
χ = C/(T-θc)
Applications for ferromagnetic materials.
Electromagnets
Motors
Transformers
Generators
What is a soft magnetic material?
A material with a low remnent magnitisation.
Small hysteresis loop.
Iron
Applications of soft ferromagnets.
Electromagnet Transformer core (response to applied field)
What is a hard magnetic material?
A material with a large remanent magnitisation.
Applications of hard magnetic materials.
Permanent magnet applications such as magnetic hard disks where bits are magnetised such that the domain can be pointing in either direction (±Mr) and retain the magentisation so it can be read.
First Maxwell equation is based on which law?
Faraday’s law.
Derive the first Maxwell Equation
Check derivation
Second Maxwell equation is based on which law?
Gauss’s Law
Derive the second Maxwell equation
Check derivation
Third Maxwell equation is based on what relation?
Magnetic flux through a surface equals zero.
How to resolve Ampere’s Law paradox for a capacitor.
Replace I with sum of conduction and displacement currents.
Fourth Maxwell equation is based on which law?
Ampere’s Law
Write fourth Maxwell Equation
Check.
What is the speed of light in terms of constants?
c=sqrt(μ0ε0)
Speed of light in medium with εr and μr?
c’=sqrt(μrμ0εrε0)
Refractive index in a medium in terms of εr and μr?
n = c/c’ = sqrt(μrεr)
If a metamaterials was to be engineered to have a negative εr and negative μr leading to a negative refractive index, what properties would it have?
Reflection rather than transmission of light casuing invisibility.
Skin depth equation for a conductor in an electric field.
E = E0 exp(-z/δ)
where
δ is skin depth
z is distance of charge from surface
Skin depth for a good conductor
δ = sqrt(2/(ωμσ))
ω is frequency
μ is magnetic permeability
σ is conductivity
