Week 1 Flashcards
What is the gradient?
Of a scalar field
Is a vector field representing slope and direction of steepest ascent of scalar field
What is divergence
Of a vector field
- a scalar field that represents whether the flow of the field is a source (positive) or a sink (negative) of field lines
What is a solenoidal field?
Divergence = 0
What is the curl?
Of a vector fields
Is a vector field that represents the direction of angular momentum vector and the speed with which an imaginary “windmill” would rotate when placed in the vector field
What is an irrotational field?
Curl = 0
Divergence theorem
The flux of a vector field out of a closed surface enclosing a volume is given by the integral of its divergence over the volume.
(Volume integral of divergence of V = Closed surface integral.da)
What does divergence represent?
The flux out of a differential volume
What is divergence theorem formula?
Triple integral over V of Nabla dot v d Tau = Double surface integral of V dot dA
What is the other name for Stoke’s theorem?
Curl theorem
What is Stokes theorem?
The circulation of a vector field V across a closed path P enclosing a surface S is given by the flux integral of its curl around a differential loop normal to that direction
- Double (surface) integral of curl of V .da = Closed loop integral of V.dl
What does the flux of the curl represent?
The total amount of swirl
How to picture 1D Dirac delta?
Infinitely high, infinitesimally narrow “spike” with area 1
Distribution not a function
What is the delta sieving property formula?
Integral (-ve infinity to infinity) of f(x) delta (x-a) dx = f (a)
How to derive dimensions of delta function?
- Inverse dimensions of its argument x
- If x is its length, delta (x) is “per unit length” or “1/length
What are the properties of 3 D delta?
Triple integral over all space delta^3 (r) d Tau =1
Triple integral over all space f(r) delta ^3 (r-a) d Tau = f(a)
What is volumetric charge density?
A scalar function which gives at every point in space the local density of electric charge. (Units of charge per unit volume.)
What is the formula for total charge in a volume?
Triple volume integral Rho (r) d Tau
How would you write point charge q at the origin in delta form?
Rho (r) = q delta (x) delta (y) delta (z) = q delta^3 (r)
How would you write an infinitesimally thin line charge distribution in delta spread along x axis?
Rho (r) = lambda (x) delta (y) delta (z)
How would you write an infinitesimally thin surface charge distribution in delta on the XY plane?
Z = 0
Rho (r) = sigma (x,y) delta z
What are the units of delta for 1-3 dimensions?
- 1: inverse length
- 2: inverse area
- 3: Inverse volume
What are the 2 paradoxes of divergence of inverse square law vector fields?
- At every location field is directed outwards yet divergence = 0
- Gauss’ theorem gives a divergence of 4 Pi
How are the paradoxes of divergence of a radial inverse square law vector field solved?
Dirac delta - infinite divergence at origin and zero elsewhere
What are the two paradoxes of the curl of an azimuthal vector field that decays linearly away from the axis?
- Curl is zero everywhere despite intuition that this should be z directed (e.g. magnetic field of straight wire)
- Stokes’ theorem suggests integral sum of flux of the curl over any surface cutting Z axis should be 2 Pi
How are the paradoxes of the curl of an azimuthal vector field that decays linearly away from the axis solved?
2D Dirac delta
What is the potential?
the negative gradient of a scalar fields
- high potential flows towards low potential
What is notable about a potential field?
Path independent
What characteristics of a field derived from a potential?
Curl is zero - irrotational
What characteristics does a curl of gradient have?
Always zero
What is a solenoidal field?
Divergence = zero
What can be done with a solenoidal field?
Written as curl of some vector function
Why can a solenoidal field be written as the curl of some vector function?
Mathematically the divergence of a curl is zero
What is the Helmholtz decomposition?
Any vector field can be written as sum of irrotational ( gradient of a scalar field) and a solenoidal field (curl of a vector field)
What is the vector triple product?
A x(B x C) = B (A.C) - C(A.B)
Symbol for permeability of free space?
Mu o
Value of permeability of free space?
4 Pi X 10^-7 N/A^2
Symbol for permittivity of free space?
Epsilon o
Value of permittivity of free space?
8.854 * 10^-12 F/m
Conceptual divergence theorem (Paco)
The net number of field lines born(divergence) minus the number of lines that die inside a volume must exit it’s surface
Conceptual Stoke’s theorem (Paco)
Multiple mini surface circular (differential curls). All internal will cancel leaving only ones at the surface
