Week 8 Flashcards
What does Faraday’s law tell us about the electric field in non-static case?
No longer conservative
What does Gauss’ law tell us about the electric field in non-static case?
B is divergencelss
What is a vector potential?
A vector field whose curl is another vector field
What is the magnetic field from potentials?
B = Curl of A (A= vector potential)
What is a scalar potential?
Where the difference in potential depends only on the different positions
What is a conservative field?
- Curl = 0
- Can be written as gradient of a scalar potential
What are gauge transformations?
Transformation of potentials that don’t change physical observables (E and B fields)
What is gauge symmetry?
The ability to change the potentials without affecting the physical observables reflects symmetry in the equations
What does gauge theory allow us to do?
Express fields in terms of potentials (but potentials are no uniquely defined)
What happens in electrostatics when you add a constant to the scalar potential, V?
The resulting field, being the gradient of V, is unchanged
In magnetostatics what happens when you add to A a term whose curl is zero?
The field whose curl is defined by A remains unchanged
What is gauge freedom?
Freedom to redefine the fields in a way that doesn’t change the physics
How to rewrite something that is conservative e.g XY?
Is XY is conservative, can rewrite as XY = -Nabla V (where V is a scalar potential)
Helmholtz theorem
Any sufficiently continuous vector field can be written as the sum of the gradient of a scalar potential + the curl of a vector potential
What is the electric field from potentials?
E = -Nabla V - PArtial derivative of A WRT t
When do electrodynamic equation become static?
When vector potential is constant in time
What if a field is irrotational?
It can be written as the gradient of some scalar field
E.g Curl of C =0; C = nabla D
What are equations for gauge transformations in EM potentials where A is a vector potential and V is a scalar potential?
A’ = A + Nabla D
V’ = V - partial derivative of D wrt t
How to use Gauge transformations with scalar functions?
- Add Nabla lambda to A
- subtract partial derivative of lambda wrt t from V
How are experimental predictions affected by gauge transformations?
They are not
What are the different choices of used for calculations called
Gauges
Name the experimental observables of EM
Only Lorentz force
Which gauge is used for electrodynamics?
Will vary for each case
What is the Coulomb gauge?
Divergence of A =0
When is the Coulomb Gauge used?
Electrostatic problems or solving electric field
What is the advantage of Coulomb’s’ gauge?
Turns scalar maxwells equation for potential into divergence of A =0
Laplacian of V = - rho/episilon o -> Poisson’s equation
What is Poisson’s equation?
Laplacian of V = - rho/episilon o
How to solve Poisson’s equation?
Set v=0 at infinity
What happen to vectorial maxwell potential equation in Coulomb gauge
Becomes inhomogeneous wave equation
- LHS -> like previous wave equations
- RHS -> source of the wave - including potential and current density
What is the disadvantage of the Coulomb gauge?
Vectorial maxwell potential results in inhomogenous wave equation which we don’t know how to solve yet (hard++) I.E. very hard to find A
What is the Lorentz gauge?
Divergence of A = -Mu o* Epsilon o * partial derivative of potential wrt t
What happens with Lorentz Gauge and Maxwell potential equations?
Get 2 inhomogenous wave equations
What are the solutions to the equations from Lorentz gauge?
Waves in V (potential) and A emanating from their corresponding sources and propagating away from the sources at the speed of light
What is the d’Alembertian operator?
A box squared
- 4d equivalent to spatial Laplacian operator
= partial second derivative wrt x + partial second derivative wrt y + partial second derivative wrt z - partial second derivative wrt t * 1/ (c^2)
What is the equation for electrostatics?
Poisson’s: Laplacian of V = - rho/ epsilon o
What is the equation for magnetostatics?
Laplacian of A = - mu o * J
What are retarded potentials?
The adjustment made for the time it takes for a change in E or B field to propagate through space.
What is retarded time?
The time at which an event or signal must have left the source point to reach the observation points at a time t, using relativistic effects
What is the formula for retarded time?
T sub r = t - (curly r/c)
What are the retarded potentials in Lorentz gauge?
The same as the static expressions with retarded time added
What is the difference between the statics equations and retarded potentials Lorentz gauge?
- retarded time in retarded potentials due to finite speed of propagation
- static case assumes changes in field propagate instantly
What is complicated about the integrals for retarded potentials?
Each r’ point will have a different retarded time. Most distant points of charge distribution have earlier retarded times that closer ones.
How does distance within the charge distribution affect retarded time?
- Further points have earlier retarded times
- Closer points have later ones
What are Jefimenko’s Equations?
A set of equations that provide a solution to Maxwell’s equations in retarded time in the presence of arbitrary current and charge distributions - general dynamic equations which include biot-savart/coulomb plus 2 additional terms
Why are Jefimenko’s equations useful?
- account for finite speed of EM wave propagation
- can calculate Em fields from time varying J and Rho
- Provide microscopic description of how each element of charge and current distribution contributes to files
How to derive Jefimenko’s equations?
Substitute retarded potential equations into definitions of the fields in terms of potentials
What does Coulomb law assume?
Charges are not moving
What does Biot-Savart Law assume?
Currents are constant and not changing with time
How does the new terms in Jefimenko’s propagate?
1/r and depend on time derivatives of J and Rho corresponding to radiation
What are Liénard-Witcher Potentials?
Generalisation of Coulomb and Biot-Savart - describe EM fields generated by moving particles
- important for understanding radiation emitted by accelerating charges E.g. EM waves or synchrotron
What are the key concepts of Lienard-Wacht potentials?
- provide way to calculate em field of particle moving with velocity v
- account for retardation effect
Formula for retarded velocity
(R(t) - r’(tsub r) )/ d sub r
What is the retardation effect?
Fields at an observation point depend on the position and velocity of the charge at an earlier time (retarded time)
What is the retarded position?
The position at which the at which the particle was when it emitted the information reaching the observation point now
What does the subscript ret in LW potentials mean?
Evaluate everything inside the braackets at the retarded time
What does it mean to take a point charge as the limit of extended charge?
To consider the process of taking a charge distribution that has a finite size and shrinking its size to zero (point charge) while keeping the total charge constant
What is an extended particle
An object that unlike a point particle has a finite size or spatial extent
What does the retardation add to an extended particle?
A factor (1 - curly r hat . (V/c) ) ^-1
What is curly r sub ret?
- r - rsub o (t sub r)
- the separation vector between the retarded position and the observation point
What is the length of the separation vector?
The magnitude of curly r sub ret
- or curly r sub ret = c ( t- t sub r)
What is v sub ret?
The velocity of the particle at the retarded time
What do retarded potentials assume about the particle?
It’s at the retarded position
How many retarded positions are there for each observation point?
One unique as long as v<c
What is the light-cone?
The region or boundary in space-time that can be reached by light or other massless particles
- events within a light-cone have causality, those outside do not
How are LW potentials related to retarded potentials?
They build on retarded potentials including velocity and acceleration of the charge
- important as accelerated charge produce radiation
What formula relates the LW potentials to each other?
A (r,t) = ( Velocity sub ret / C^2 ) * scalar potential. (R.t)
What is the acceleration of the charge at the retarded time
Time derivative of V sub ret = (R(t) - r’(tsub r) )/ d sub r^2
What is the condition for using retarded potentials?
Magnitude of X must be <ct
How to obtain electric field from vector potential equation?
Negative time derivative of potential
What happens with equations for E fields for a moving point charge in static case?
Falls off as inverse square of distance to particles
- becomes static equation as v and a go to zero
What happens with equations for B fields for a moving point charge in static case?
First term falls offs inverse power or curly r and is the dominant field at large distances.
Which term is responsible for EM radiation?
The second term (B field) in equations for fields in moving point charges
What term is proportional to retarded acceleration?
The B field equation for moving point charge
What is EM radiation proportional to?
1/curly r
What is a generalised Coulomb field?
The E field equation for moving point charge as becomes static equation if v and a =0
What does curl of magnetic vector potential produce?
Magnetic field
What does the time derivative of the magnetic vector potential produce?
Electric field
How to obtain magnetic field from vector potential equation?
Curl of vector potential
How do you prove gauge freedoms?
Substitute gauge transformations into equations for B and E (i.E curl of A = B)
What is formula of Coulomb gauge?
Divergence of A =0
What are the most general sources of EM?
Charge and current density
How does velocity affect charge size?
Approaching charges look bigger, departing look smaller
What is intuition for life cone shape?
Retarded time equation is equation of a cone
Which gauge is used for magnetostatics?
divergence of A = 0
