PX157 - B5c - electric field from an infinite line of charge

λ = \frac{Δ Q}{Δ L} \to c o n s t a n t

step 1: symmetry
- $\vec{E}$ points radially outwards in a cylindrical sense (away from the axis), ie: $\vec{E} = E \hat{R}$
- $| \vec{E} |$ is constant on a cylindrical shell concentric with the line of charge
step 2: gaussian surface
- cylinder of radius, $R$ , and length, $L$
- key point: the cylindrical shell is still open, two circles are needed at the top and the bottom to create an enclosed shell
- top:
$\vec{E} = E \hat{R}$
$d \vec{S} = d S \hat{z}$
$\vec{E} \cdot d \vec{S} = E d S \hat{R} \hat{z} = 0$
- bottom:
$\vec{E} = E \hat{R}$
$d \vec{S} = - d S \hat{z}$
$\vec{E} \cdot d \vec{S} = - E d S \hat{R} \hat{z} = 0$
- curved:
$\vec{E} = E \hat{R}$
$d \vec{S} = d S \hat{R}$
$\vec{E} \cdot d \vec{S} = - E d S \hat{R} \hat{R} = E d S$

∴ \subset \supset \iint \vec{E} \cdot d \vec{S} = \iint_{c u r v e d} E d S = E 2 π R L

Q_{e n c l .} = λ L

E 2 π R L = λ \frac{L}{ϵ_{0}} ⟹ \vec{E} = \frac{λ}{2 π ϵ_{0} R} \hat{R}