Week 2 Flashcards
What is retardation?
Delay in the movement of charge being felt by another
What does curly r represent?
Separation vector: r - r’
What is the net force acting on a charge in words?
The vector sum of the forces caused by every other charge in the universe
What is force in terms of electric fields?
F = QE
How do equations change for considering continuous distribution instead of point charge?
Integration
Formula for electric fields of continuous charge distribution
E ( r ) = (1/ (4 PI Eo) * volume integral ( rho (r’) / curly r^2 ) * curly r hat d Tau’
Formula for electric field of point charge
AKA coulomb law
E(r) = 1/ (4piEo) * q/ curly r^2 * curly r hat
What is dq for a line?
λdl
What is dq for surface?
σda
What is dq for a volume?
ρ d τ
How to turn general case into specific dimension?
Dirac delta
Where do electrostatic lines start and finish?
Start as positive, end at negative or infinity
How does is flux arranged in flux tubes?
Constant through every cross-section of the tube
By what proportion does the CSA of flux tubes grow?
R^2
By what proportion does flux density in tubes change and why?
1/r^2 to compensate for increasing area
What are inward flowing flux tubes?
Negative flux
What are outward flowing flux tubes?
positive
What is net flux through a closed surface proportional to?
Total charge( positive - negative) inside the volume enclosed by the surface
- Intregral form of Gauss
How to obtain differential form of Gauss from integral?
Sub in:
- RHS: Divergence theorem -> Flux = Volume integral of divergence of field
- LHS: Q enc = volume integral of ρ d τ
What does Gauss law in differential form tell us?
Divergence of electric field is proportional to the volumetric charge density at every point in space
What is the Gaussian surface for spherical symmetry?
Concentric sphere of varying radius
What is the Gaussian surface for cylindrical symmetry?
Coaxial cylinder of varying radius s, arbitrary length l
What is the Gaussian surface for plane symmetry?
Gaussian pillbox (rectangular cuboid) that straddles the surface
E.da = [Ez Z1 - Ez Z2]
When does Gauss become less useful?
When symmetry is broken
What is special about an irrotational field?
It can be written as the gradient of a scalar potential field because it derives from a scalar potential (V)
Which 2 statements are implied from an irrotational field?
E = - ∇ V
V (b) - V (a) = - ∫ E . Dl
What is the reference point for V (r)?
Technically a difference between 2 points
- typically set r to infinity so V(r) = 0
What is the formula for potential of a point charge?
V = 1/ (4 * pi * Eo) = q/ curly r
What is r?
Vector from origin to source
What is r’?
Vector from origin to test charge
What is electric field definition?
Force per unit charge as if a test charge was placed at that point
What is superposition?
When forces/fields are added vectorial
How to depict Dirac delta for a plane?
E(r) = ∫ ∫ 1/ (4 * pi* Eo) * ρ d T / (curly r^2) * curly r hat
E(r) = ∫ ∫ 1/ (4 * pi* Eo) * σ δ(z) dxdydz / (curly r^2) * curly r hat
E (r) = 1/ (4 * pi* Eo) * ∫ δ(z) dz ∫ ∫ σ dxdydz / (curly r^2) * curly r hat
- where δ(z) dz = 1
What is the work to move test charge?
V (b) - V (a) = W/Q
What is the work to assemble charge distribution
W = Eo/2 ∫ ∫ ∫ [E] ^2 d τ
Units for potential
NM/C or J/C
Name and value for Eo
Permittivity of free space = 8.85 * 10 ^-12 c^2/ NM^2
What is the sign convention for potential and charges?
Positive for positive charge as work is needed to bring test charge closer
What is the work bringing an isolated charge from infinity to isolated area/
0 as no other charges exist to exert a force
What is the total work to move numerous charges?
Qi (1st charge moved ) * sum of potential for each individual charge
What does work to assemble configuration of point charges represent?
Energy stored in configuration, amount of work you would get back on dismantling the system
What is the formula for energy of a continuous charge distribution?
W = 1/2 ∫ ∫ ∫ ρV d τ
= Eo/2 ∫ ∫ ∫ [E]^2 d τ —> form substitution Gauss Law and ∇ V = -E
What is a dipole?
Two equal and opposite charges separated by vector d
What is the direction of vector d in dipole?
Negative to positive
What is the formula for superposition of dipole
V (r) = 1/ (4 * PI * Eo) (q/r+ - q/r-) = q/(4 *pi * Eo) * (r- - r+)/(r+r-)
What happens to V dipole at long distances? I.e r»d
- r- -r+ ~ d cos θ and r+r- ~ r^2
V (r) = 1/ (4 * PI * Eo) (q d cos θ) / r^2
Where r= curly r
What is the vector dipole moment?
P = q d = q d dhat
V (r) = 1/ (4 * PI * Eo) (r hat . P) / r^2
What is the difference between potentials of a dipole compared to point charge?
- Dipole decreases by 1/r^2
- Point charge decreases by 1/r
How to arrive at idealised point dipole?
decrease distance d to 0 and increase q to infinity to keep p constant
p = qd
When do Q and d cease to be important?
When d«<r
When does dipole become relevant?
When monopole term = 0
Total charge often 0 as positive and negative terms like to pair up
- if monopole = 0 , dipole dominates
What is the dipole moment for a collection of charges?
P = ∑ ri . Qi
What is the dipole for charge distribution?
P = ∫ ∫ ∫ r ρ d τ
What s the charge for a collection of point charges in a momopole?
Q tot = ∑ qi
What is the charge for charge distribution in a monopole?
Q total = ∫ ∫ ∫ ρ d τ
How to derive equation for physical dipole with 2 charges?
P = ∑ ri . Qi
- Diagram showing r1-r2 = d, where q1=-q2
P = r1q1 + r2q2
= q (r1 - r2)
= qd
