Week 6 Flashcards
What is the formula for force per unit charge that can act on a charge inside a circuit?
F = Fsource + E + vxB
What is Fsource?
- Only nonzero inside a source and essentially Lorentz force
- Battery -> chemical force
- Thermocouple -> temperature gradient
- Photoelectric cell -> light
- Van de Graff generator -> electrons onto conveyor belt
What is the difference between Lorentz force inside and outside source?
- Inside force only experienced inside and can be ignored for calculations outside
Define EMF
- Closed loop integral of f.dl
- Change in energy over change in charge
- Work per unit charge
What are the parts in the EMF integral?
- dL - path of circuit/charge takes
How to use EMF integral to find F source?
If assume static circuit (E.dl = 0) and resistive wire , all terms in Lorentz equation cancel except Fsource.
E = Closed loop integral Fsource.dl
Which way around the circuit does F source drive current?
Opposite to electric field
What is the function of a battery?
To maintain a a voltage difference equal to the EMF
How can non-zero EM arise without a battery?
- From magnetic term when the circuit moves (notional EMF) - closed loop integral of (v x B).dl does not = 0
- From electric term when E field is no longer electrostatic - Closed loop integral of E.dl does not equal zero
What is an analogy for F source?
In gravitational potential, the resistive force against gravity
Which direction is electric field in a circuit?
From positive to negative
What is F source in ideal resistance-less batteries?
-E field
What is F source in batteries with internal resistance?
Force needs to > E field,
- Force = F source + E field = J/sigma
How to magentic forces contribute to EMF in circuit?
- Zero as perpendicular to path C
What is motional EMF?
Voltage induced when a conductor moves through a magnetic field
If a loop of wire is pulled partly into a magentic field, what happens?
No F source and No E field
- only magnetic term left and appears to do work
What is the formula for EMF in a rectangular loop moving in a B field?
EMF = vBh where h is the width of the wire traversing the field (face on)
OR
EMF = -D Flux/dt
How to find EMF from magnetic flex?
Flux = Double integral of B . Da = Bhx (essentially Bxy)
D Flux/ dt = Bh (Dx/dt) - changing variable = - Bhv
What is the formula for work done by a magnetic field?
Always = 0 according to Maxwell’s Equations
What is the Universal Flux rule?
Whenever (and for whatever reason) the magnetic flux through a loop changes, an emf will appear in the loop
EMF = -D flux /dt
What is Lenz’s Law in words?
Nature abhors a change in flux
Why is the EMF in a loop moving through a magnetic field negative?
It is an induced EMF working to oppose the change in flux
What is the equation for mutual inductance?
Flux 2 = M21 * Current 1
Where M is the constant of propotionality
What does mutual inductance depend on?
- Physical geometry of loops
- Orientation of loops
- Medium between loops
How is M21 different to M12 and what is the consequence of this?
M21=M12
Flux produced on loop 2 by current in loop 1 = flux produced on loop1 by current in loop 2
What is self inductance?
When a loop of wire is affected but the magnetic field it creates due to current flowing through it
What is the formula for self inductance?
FLux = L*I (L = proportionality constant, inductance)
What is the effect of induced EMFin a loop with no current?
To try to keep current at zero
What is the equation for energy stored in a current on any loop of wire?
W (work done) = 1/2 L I^2
What is the equation for energy stored in a magnetic field?
W (work done) - 1/ (2Muo) Triple integral of magnitude of B field^2 D Tau
(Volume integral)
How do you calculate work done by a magnetic field?
This is always zero
- displacement perpendicular to force
Explain inconsistency in Ampere’s Law with the continuity equation
Continuity equation states Nabla. J = - Dp/Dt
-> any net flux of current into a volume means that charge is accumulation inside the volume as time progresses
Ampere’s Law: Nable x B = Muo J
-> If take divergence, 0= Muo Nabla.J
**If static case,no time derivatives so consistent
What is the divergence of curl?
Zero
Explain the inconsistency in Ampere’s Law for a charging capacitor
Integral form of Amperer’s is integral of B.dl
But you can choose different surfaces enclosed by same loop and have different currents
How did Maxwell solve inconsisitencies with Ampere’s Law?
By adding a new term (MuoEpsilon o DEfield/Dt)
When is Maxwell’s new term for Ampere’s Law required?
With time varying electric fields
What is a displacement field?
With time varying electric fields
What is the integral form of Maxwell’s correction to Ampere’s Law?
Closed loop integral of B.dl = Mu o ( Ienc + Epsilono dE Field Flux/Dt)
What is a displacement current?
A Quantity that represents the rate of change of the electric field where there is no physical current but where the electric field changes over time
What is the formula for displacement current?
Jd = Epsilon 0 DE field/
How does a displacement current arise?
From a time varying electric field without real charge flow (Current) present
What did Maxwell’s correction to Ampere’s Law do?
Connected E and Bfields predicting EM
What is Gauss Law (Electric Fields) as Maxwell’s law?
-Divergence of E field= charge density (Rho) / epsilon o
Electric charges create electric fields
What is Gauss Law (Magnetic Fields) as Maxwell’s law?
Divergence of B fields = 0
Magnetic fields always form loops and never have starting or ending points (no monopoles)
What is Faraday’s Law as Maxwell’s law?
Curl of E field = -DB/dt
Changing magnetic fields create E fields
What is Ampere’s Law as Maxwell’s law?
Curl of B = Muo( J + epsilon o (dE field/dt).
(J= current density)
Electric currents and changing electric fields create magnetic fields
In which situations are Maxwells equations most useful?
In a vacuum
What are auxiliary fields?
Mathematical tools to separate effects of materials from the fields created by charges and currents i.e. make it easier to describe and calculate how E and B fields behave in materials when those materials react strongly to those materials
What is the purpose of the electric displacement field?
Separates the effect of free charges from the material’s internal response to the electric field
What is the purpose of the electric displacement field?
Separates the effect of free charges from the material’s internal response to the electric field
E.g. a displacement field would explain how much water (electric charge from E field) a sponge (the material) can hold, plus how the sponge reacts to the polarisation
What is the symbol for Electric displacement field?
D
What is the symbol for magnetic field intensity?
H
What is the purpose of magentic field intensity?
Helps us understand how magnetic fields behave in materials, especially those which can become magnetised
E.g. H field tells us how much of a magnetic field comes from an external magnet and how much comes from the rod’s own magnetisation
What is the formula for an electric displacement field?
D = Epislon o E + P
P = polarisation of material
What is the formula for magnetic field intensity?
H = 1/ Muo*B - M
Where M = magnetisation
What is the polarisation current?
A current density which appears in addition to the bound current when considering auxiliary fields in static cases.
It is not a real current but explains for E fields interact with materials that don’t conduct
Describes changes in materials polarisation over time and arise due to shifting of bound currents
What is the formula for a bound current?
Jb = Curl of M
What is a bound current?
Current s that arise in magnetic materials due to magnetisation of the material
Current s that arise in magnetic materials due to magnetisation of the material
What is the formula for bound surface current density?
Kb = M x (cross) unit normal vector
M = magnetisation vector)
What is the symbol for bound surface current?
Kb
What is bound volume current?
Currents which occur inside a material as the result of the curl of the magnetisation within the material
What is the formula for bound volume current density?
Jb = Curl of M
(M = magnetisation vector)
What are free currents?
Currents due to the movement of free charges
What is the formula for total bound charge in a general dynamic case?
Rho sub b = - Divergence of p (polarisation vector)
What is the formula for total bound current in a general dynamic case?
Jb = Curl of M + Dp/Dt
What does the total charge and current consist of?
The total free plus bound current and charge
What is the formula for total charge?
Rho = Rho sub f + Rho sub b
= Rho - Divergence of P
What is the formula for total current?
J = Jf + Jb = Jf + Curl of M + dP/dt
What is the effect of substituting equations for total charge and current in Maxwells equations?
There. Is no mention of bound charges, current, polarisation nor magnetisation.
Only auxiliary fields and free charges and currents remain
What is the macroscopic Gauss Law (Electric Fields) as Maxwell’s law?
Divergence of electric displacement field = Rho sub f
What is the macroscopic Gauss Law (Magnetic Fields) as Maxwell’s law?
Divergence of B = 0
What is the macroscopic Faraday’s Law as Maxwell’s law?
Curl of E field = - dB/dt
What is the macroscopic Ampere’s Law as Maxwell’s law?
Curl of H = Jf + dD/dt
What is the difference between microscopic and macroscopic Maxwell’s equations?
Micro: focus on fields generated by free charges and currents
Macro: considers materials response and all you need to know is how M and P depend on E and H
What are constitutive relations of the material?
How materials respond to E and H fields
Why do constitutive relations matter?
They tell us how the material affects the electric and magnetic fields - D and E are directly related in a vacuum but this changes in a material
They link the fields to the material properties
What are the formulas for constitutive relations in linear materials?
- P = Epsilon o Chi m E field
(Chi m = funny looking X, magnetic susceptibility)
- M = Chi m H field
What is the formula for the electric displacement field in linear materials?
D = Epsilon E field
What is the formula for the magnetic field intensity in linear materials?
H = 1/Mu * B
What is the formula for electric permittivity in a linear material?
Epsilon = Epsilon 0 Epsilon r = Epsilon o (1 + Chi sub e)
What is the formula for magnetic permeability in a linear material?
Mu = Mu o Mu r = Mu o (1 + Chi m)
What do boundary conditions do?
Govern how fields will behave at interfaces of materials
What are the boundary conditions in Maxwell’s equations?
Parallel:
- E1- E2 = 0
- H1 - H2 = Kf x (cross) unit normal vector
Perpendicular:
- B1 - B2 = 0
- D1 - D2 = Sigma sub f
Describe the boundary conditions in Maxwell’s equations for E fields
Tangential - continuous
Normal - discontinuous
- The tangential component is continuous across a boundary
- the E field parallel to the surface does not create a jump or a discontinuity unless there is a surface charge - The normal component of E is discontinuous if there is a surface charge density
- if there is a surface charge, the normal component will change across the boundary (push/pull the E field)
Describe the boundary conditions in Maxwell’s equations for D fields
Tangential - Continuous
Normal - Discontinuous
- Tangential component is continuous
- because D takes into account polarisation of the material which reacts smoothly to changes in property of material - Normal component of D may experience a discontinuity due to a free surface charge
Describe the boundary conditions in Maxwell’s equations for B fields
Tangential - continuous
Normal - continuous
- Tangential component of B is continuous unless there is a surface current
- because linked to magnetic permeability and magnetic field lines which remain continuous - Normal component as there are no magnetic monopoles
Describe the boundary conditions in Maxwell’s equations for H fields
Tangential - Discontinuous
Normal - Continuous
- Tangential component is discontinuous in the presence of a surface current
- the tangential magnetic field will jump across the boundary - Normal component is continuous
- there are no magnetic charges so H does note change abruptly
What is the general trend of tangential vs normal boundary conditions?
Tangential generally smooth and continuous as less affected by surface charges and currents, whereas normal more strongly influenced with more discontinuities
What’s is the symbol for bound volume current density?
Jb
Currents due to the movement of free charges
