Week 6 Flashcards
Magnetic part of Lorentz force law
Magnetism analogue of Coloumb’s law
Which is force on test charge Q, moving with velocity v in magnetic field B
Why are the velocities in magnetostatics?
The movement of charges causes a static magnetic field (unchanging magnetic field)
Right hand rule
For cross product eg V x B
First point thumb in direction of V, then index finger in direction of B
Resultantly middle finger points in direction of cross prod vector
Each subsequent finger is orthogonal
Lorentz force law
Prove that the force on a closed loop of wire carrying current I due to a constant magnetic field B is zero
The size of force due to a magnetic field is?
Where α is the angle in between v and B
The field B produces no force on the static test charge Q
Therefore the size of force depends only on |v|Sinα
Interesting consequence of force law
Direction of force depends on sign of charge
You can tell the charge of a particle by seeing which direction it is deflected in when moving through a magnetic field
Magnetic forces and work
Fmagnetic is perpendicular to v therefore force due to magnetic field does not do any work as
Fmagnetic • dl = Fmagnetic • vdt = 0
Magnetic forces can’t accelerate a charged particle, only alter direction that test charges move
A current is
A charge in motion
Units for current
1 Ampere = 1 coulomb per second
Current vector points
In the direction opposite to the movement of electrons
Calculate Fmagnetic for charge density λ and on a length of wire
Define the magnetic force on a surface
Define a magnetic force through a volume
Derive the continuity equation
Taking the volume current density (which ishow much charge flows through a surface area element):
J • da = ρvCos(θ)da
Which then in time is ρ(vCos(θ)dt)da then
Steady current
No build up of charge
Assumptions of electrostatics and magnetostatics
accordingly
stationary charge
steady current
Significance of?
This is a consequence of working with magnetostatics
This means the field lines of J can’t begin or end on any local point (compare to grad • E = ρ/ε0 in electro case where electric field lines can begin or end on any charges which give rise to nonzero value of ρ)
Biot Savart rule
Physical rule that governs how steady currents in wires produce static magnetic fields
Where:
I = Idl’ for a current wire
P is path along wire
s = r - r’
μ0 ?
4π * 10**-7
Right hand rule for magnetic field
If you wrap your hand around a wire such that your thumb is pointing in the direction of flow of the current through that wire (up or down) then
The direction that your fingers curl around the wire represent that direction of flow of the magnetic field at some arbitrarily close distance
Trig substitutions
Sin rule
Cosine rule
Angular basis vector in cylindrical coordinates
Find magnetic field at a fixed distance ρ from an infinite straight wire on the z axis carrying steady current I travelling up in z direction
Where r is a point at radial distance ρ from the wire
When does a charge NOT contribute to the flux integral of an electric field
When the charge is located outside of the surface being integrated over
When does a current not contribute to the line integral of a closed path
When the path doesn’t enclose the wire
Show the curl of B around the infinitely long straight wire coinciding with z axis
Deduce that
Given that rhs of stokes theorem = μ0*I P
State Biot Savart law in terms of volume current density J
Use Biot Savart to deduce
Derive Ampere’s law
Magnetic field as of an infinite charged plane
Where K is the uniform surface current
Motivate the vector potential
Given M3, it would be sufficient to show that B can be given as a curl of a vector field
This is because the divergence of the curl = 0
Find Kronecker deltas from epsilon tensors
Laplacian in components
Is magnetic potential uniquely defined
No, it is defined up to a gradient (as opposed to constant for electro)
Derive the magneto equivalent of poisons equation with solution
Deduce vector potential for B produced by an infinitely long wire carrying charge I = Iez and coinciding with z
Derive the first 2 terms of multiple expansion for magnetic field
