Week 3 Flashcards
How are conduction electrons positioned?
- Sea surrounding positive ions
- Positioned to ensure electric field inside conductor is zero at every point
What do we assume about conductors?
Perfect with unlimited supply of free charge
When is force exerted on charges
With unlimited free charge until static equilibrium reached
What is the electric field at equilibrium in a perfect conductor?
Net field inside = 0
What is charge density inside a conductor and why?
ρ = 0 from Gauss’ Law
Where do net charges reside in a conductor and why?
On the surface
- Gauss law means they create a field which is felt outside and cancels net internal E field
What is the potential for conductors and why?
V(r) = constant
Any path through inside will have E=0along all points
What is the orientation of the E field?
Perpendicular (tangential would mean charges move along surface)
Define floating conductors
Isolated, not connected to anything so charge cannot flow in or out of them
What is the value and distribution of charge in floating conductors?
- Fixed total net charge as cannot flow in or out of them ( neutral (Q=0) or isolated)
- can be redistributed to maintain E = 0
What is the potential in a floating conductor?
Variable depending on changes in charge distribution to maintain E = 0
What is the value and distribution of charge in fixed conductors?
- Connected to potential source +/- earthed or grounded
- Varies as need to maintain fixed potential and E = 0 inside
Define induced charge
A charge that appears on a surface when a charge is moved closed to a conductor to maintain equilibrium
How do induced charges differ in grounded and floating conductors?
Floating = if negative is induced on one surface, positive induced on opposite so total net charge =0
Grounded = if negative is induced on one surface, voltage source can correct for this
What is the criteria for Gaussian surface inside conductors?
Must have zero flux so it has zero net charge inside
What is the electric field vector?
E (r) = σ (r) / εo
Differential potential equation
V = - ∫ E.dl
Define capacitance
Constant of proportionality between charge and potential difference
C = Q/V
What does capacitance depend on?
Geometry of conductors (size, shape, separation)
What are the types of induced dipoles in matter
- Atomic polarisation
- ionic polarisation
- Molecule polarisation
Describe atomic polarisation
- Neutral atoms
- Electric field creates a polarisation (dipol
- Electron cloud displacede) and becomes a conductor
Formula for induced dipole
P= α E
α = atomic polarisability
What does α depend on?
Detailed structure of atom
Also α = εo χo ( perm vs electric susceptibility)
What makes up dipole forces?
- Electric field pushes apart
- Mutual attraction of dipoles pulls together
Describe ionic polarisation
- For dielectrics made of lattice of ions
- Lattice displaced
- External electric field -> small displacement between sublattices causing electric dipoles
Describe molecule polarisation
- Polar molecules aligned
- Have pre-existing permanent dipoles (polar) but net moment usually zero
- In electric field positive and negative experience different forces creating torque to align dipole to E
Formula for permanent dipole
P = qd
Formula for torque from external field on permanent dipole
N = p cross E
What causes a dielectric to become polarised?
Applied electric field
What is the dipole moment per unit volume?
P (r) = n . P
(N molecules per unit volume each with a dipole p)
What are bound charges?
Those bound to an atom or molecule and unable to move freely
Formula for surface bound charges
σ = P.n
- maximal when parallel, 0 when orthogonal
What is the orientation of uniform surface bound charges?
- 2 opposite edges polarised due to accumulation of charge
- p = 0 centrally
- n hat points out from other edges
Formula for volumetric bound charges
ρ = - ∇. P
What is the orientation for non-uniform volumetric bound charges
Radial, with negative accumulation at centre
Formula for total charge
ρ = ρ free + ρ bound
- free = controlled in experiement e.g. set potential ( ∇ .E = ρfree/ εo)
- bound = appear in dielectrics due to polarisation
Derivation of macroscopic Gauss Law equation
- Gauss Law
- Sub in ρ = ρf + ρb
- Sub in ρb = - ∇.P
- Rearrange for ρf
5.Factor out ∇
- Sub in D = εoE + P
Differential form of macroscopic Gauss Law
∇.D = ρf
Integral form of macroscopic Gauss Law
∫ ∫ D.da = Qfree enc (via divergence theorem)
Why is D not εoEfree?
(May assume as ∇.Ef = ρf/ε and ∇.D = ρf
- Although same divergence as Efree, curl is different so different fields
∇ x E free = 0
∇ x D = 0 + ∇ x P
Why are displacement fields useful?
- Simplify the analysis of electric fields in materials that are not perfect conductors (like dielectrics).
- The displacement field accounts for both free charges and bound charges in a material.
- It separates the effects of free charges from the material’s response (polarization), making it easier to apply Gauss’s Law in different media.
Define electric susceptibility
measures how much a material becomes polarized in response to an applied electric field
What are linear dielectrics
- materials where the polarization is directly proportional to the applied electric field.
- the relationship between the electric field E and the polarization P is linear, meaning that the material’s ability to polarize doesn’t change with the strength of the electric field
Displacement field formula
D = εo (1 + χo) E
When do volumetric bound charges appear?
Inside volume of a dielectric when P and a non-zero divergence
How to calculate field ?wihtin a material?
once found bound charges, can ignore polarisation vector P and calculate field from bound charges and any external or free charges
What are the bund charges for. Uniformly polarised sphere, with constant polarisation P throughout
- Bound volumetric= 0 as divergenceless
- Bound surface = P. N hat
What is the electric field for 2 spheres of opposite charge, slightly offset?
E = Esph1 + Esph2
= (ρ /3εo )( r- r+) - (ρ/ 3εo) (r - r-)
= - (ρ/ 3εo) (r+ - r-) = - ρd/ 3εo
= - P/ 3εo
What does the result for electric field inside a sphere tell us
- Inside is a uniformly polarised sphere independent of position
- field produced by bound charges is constant inside p here and opposite to polarisation
- field outside corresponds to that of a physical dipole
What is relative permittivity
dielectric constant, is a measure of how much a material can store electrical energy in an electric field compared to a vacuum
= 1+ χo
How do electric and displacement fields differ at interfaces of mediums with different permittivity
- electric field E can change at the interface of different materials because it depends on the material’s permittivity
- displacement field D is continuous across the interface when there are no free surface charges.
When is D proportional to E?
Only inside linear dielectric
What is the dielectric’s effect on capacitance?
- increases the capacitance of a capacitor by a factor of εr
- because bound charges act like tiny dipoles and create their own electric field that opposes the original electric field
- so reducing the electric field inside the capacitor for a given charge
