Capacitors and E-fields

Question 1

work done by an electric field

Answer 1

W = ∫F⋅dl = q∫E⋅dl
dl and E are vectors

∫E⋅dl is called the circulation

Question 2

what is meant by ‘electric fields are conservative’

Answer 2

the work done in moving a charge from point A to point B in the field is independent of the path taken

Question 3

electric potential equations

Answer 3

ϕ = -∫E⋅dl (limits from r to 0)

E= -∇ϕ

Question 4

What is the circulation of an E-field

Answer 4

∮E⋅dl = 0

For a closed loop, the work done is 0

Question 5

properties of a conductor

Answer 5

1. perfect conductor has no resistance
2. all points in a conductor have the same ϕ value
3. E=0 in a perfect conductor
4. Any excess charge is uniformly spread on the surface (due to 3)

Question 6

capacitance equation

Answer 6

Q = C*Δϕ (or v= Q/C)
Q: charge stored
C: capacitance
Δϕ or v : potential difference

Question 7

what is a capacitor

Answer 7

a device that stores charge when connected to a potential
since Δϕ ∝ Q, the capacitance, C, is the constant of proportionality

Question 8

derive the capacitance of a parallel plate capacitor

Answer 8

E-field is from Gauss’ law an an infinite surface
E = σ/2ε₀ (then *2 since theres 2 plate)
E = σ/ε₀ = Q/ε₀A

Δϕ = ∫E dl = Qd/ε₀A (d is the plate separation)
C = Q/Δϕ = ε₀A/d

Question 9

derive the capacitance of a spherical capacitor

Answer 9

A sphere inside a hollow sphere. Inner radius is rᵢ and outer radius is rₒ

E = q/4πε₀r²
Δϕ = ∫E dr = q/4πε₀ [1/rᵢ - 1/rₒ] = q(rₒ -rᵢ)/4πε₀rᵢrₒ

C= q/Δϕ = 4πε₀rᵢrₒ/(rₒ -rᵢ)