Institutt for fysikk, NTNU TFY4155/FY1003: Elektrisitet og magnetisme Spring 2005

Summary, week 3 (January 18 and 19)

Electric field [FGT 22.1; YF 21.4; TM 21.4; AF 21.5; LHL 19.4; DJG 2.1.3]

E = \frac{F}{q}

= force pr unit charge

SI unit for electric field: [E] = N/C

Electric field due to point charge [FGT 22.1; YF 21.4; TM 21.4; AF 21.6; LHL 19.5; DJG 2.1.3]

E = \frac{q}{4πε_{0}r^{2}} r̂

The superposition principle for electric field:

E = ∑_{j=1}^{n} E_{j} = \frac{1}{4πε_{0}} ∑_{j=1}^{n} \frac{q_{j}}{r_{j}^{2}} r̂_{j}

Continuous charge distributions [FGT 21.4, 22.3; YF 21.5; TM 22.1; AF eks. 21.6; LHL 19.5; DJG 2.1.4]

On a length scale which is large compared to the distance between single charges, one ”sees” approximately a continuous charge distribution. (Analogy: Macroscopic objects have an approximately continuous mass distribution even though they are actually built up from ”discrete masses” (i.e. atoms).)

Sum over point charges is now replaced by an integral over a charge distribution:

∑_{i} Δq_{i} →^{Δq_{i}→0} ∫ dq

3D (= 3 dimensions): volume charge

dq = ρ dV
ρ = ρ(x, y, z) = charge pr unit volume = volume charge density
[ρ] = [q/V] = C/m^{3}
Volume element : dV = dx dy dz (cartesian coordinates)
= r^{2} sin θ dθ dϕ dr (spherical coordinates)
= ρ dρ dϕ dz (cylinder coordinates)

2D: surface charge

dq = σ dA
σ = σ(x, y) = charge pr unit area = surface charge density
[σ] = [q/A] = C/m^{2}
Surface element : dA = dx dy (cartesian coordinates)
= r dϕ dr (polar coordinates)

1D: line charge

dq = λ dl
λ = λ(x) = charge pr unit length = line charge density
[λ] = [q/L] = C/m
Line element : dl = dx (straight line)
= R dϕ (for circle with radius R)

Electric field in a distance r from an infinitesimal charge dq:

dE = \frac{1}{4πε_{0}} \frac{dq}{r^{2}} r̂

Electric field from continuous charge distribution:

E = \frac{1}{4πε_{0}} ∫ \frac{r̂ dq}{r^{2}} =^{3D} \frac{1}{4πε_{0}} ∫ \frac{r̂ρ dV}{r^{2}}

More precisely: The electric field E(r) in a position r = (x, y, z) due to a charge distribution described by the charge density ρ(r′) = ρ(x′, y′, z′) is given by

E(r) = \frac{1}{4πε_{0}} ∫ \frac{(r − r′)ρ(r′) dV′}{|r − r′|^{3}}

where dV′ = dx′ dy′ dz′ (in cartesian coordinates) is a volume element in position r′. Note that r does not have the same meaning in the last two equations. In the first one, r denotes the vector from dq to the point where you are supposed to find E. That means r is different for the different charge elements dq in the system we are looking at. In the second equation, r denotes the position where E is supposed to be determined, whereas r′ is the position variable for the charge density ρ. We had a choice here concerning notation: We could have introduced a new vector R ≡ r − r′ and then write R̂/R^{2}, or, as we actually did above, rewrite the unit vector. Let’s look at a figure:

We see that the unit vector in Coulombs law should point from the volume element dV′ in position r′ towards the position r. Hence, we may write (r − r′)/|r − r′|^{3} in the expression for E(r).