2024-01-22 Conductors

Science slam

• 4 minute short format presentation
• On anything EM theory
• Deadlines
  • 3/1/24 Submit favorite science slam to professor
    • Science Slam English
  • 3/22/24 pick a topic
  • 4/1/24 prepare presentation

Review

• Electric charges re the sources of electric fields
• If a field is present there must be a charge
• If curl is 0, then a field is an electrostatic field
• Introduction of potential as a scalar function of electric field
• Poisson’s Equation and Laplace’s equation

Map of solving math problems

*** insert photo here  ***

Conductors Lecture

• Conductors are materials in which electrons can move freely.

  • Metals are conductors

• This is because the valence band and conduction band overlap at a microscopic scale

• Basic facts about conductors

  • The static field and net electric charge inside a conductor is always 0
  • Any net electrical charges sit on the surface of the conductor
  • The static electric field is always perpendicular at the surface of the conductor
  • Magnitude of static electric field at surface
  $$E=\frac{\sigma }{{ϵ}_0}$$
  • If the surface is grounded, potential at surface is 0
  • Surface charge density can be calculated as (check this)
  $$\sigma =-{ϵ}_0\frac{dv}{d\tau }$$
  • The movement of charge is basically instantaneous
  • Charges distribute themselves at the surface, leading to 0 electric field

• Electrons will move freely throughout a conductor, moving towards positive charges and away from negative ones

In class questions

A metal sphere of radius R, carrying charge q, is surrounded by a thick concentric metal shell (inner radius a, outer radius b, as in Fig. 2.48). The shell carries no net charge a. Find the surface charge density σ at R, at a, and at b.

$${\sigma }_R=\frac{q}{4\pi R^2}$$ $${\sigma }_a=-\frac{q}{4\pi a^2}$$ $${\sigma }_b=\frac{q}{4\pi b^2}$$

b.

$$\frac{1}{4\pi {ϵ}_0}\left(\underset{0}{\overset{R}{\int }}0dr+\underset{R}{\overset{a}{\int }}\frac{q}{r^2}dr+\underset{a}{\overset{b}{\int }}0dr+\underset{b}{\overset{\infty }{\int }}\frac{q}{r^2}dr\right)$$ $$V=\frac{1}{4\pi {ϵ}_0}\left(-\frac{q}{a}+\frac{q}{R}-\frac{q}{b}\right)$$

c.

$${\sigma }_R^{\prime }=\frac{q}{4\pi R^2}$$ $${\sigma }_a^{\prime }=-\frac{q}{4\pi a^2}$$ $${\sigma }_b^{\prime }=0$$ $$\frac{1}{4\pi {ϵ}_0}\left(\underset{0}{\overset{R}{\int }}0dr+\underset{R}{\overset{a}{\int }}\frac{q}{r^2}dr+\underset{a}{\overset{b}{\int }}0dr+\underset{b}{\overset{\infty }{\int }}\frac{0}{r^2}dr\right)$$ $$V=\frac{1}{4\pi {ϵ}_0}\left(-\frac{q}{a}+\frac{q}{R}\right)$$

2.39 a.

$${\sigma }_a=-\frac{q_a}{4\pi a^2}$$ $${\sigma }_b=-\frac{q_b}{4\pi b^2}$$ $${\sigma }_R=\frac{q_b+q_a}{4\pi R^2}$$

b.

$$E_{\text{outside}}=\frac{1}{4\pi {ϵ}_0}\frac{q_a+q_b}{r^2}\perp \text{to outside of conductor}$$

c.

$$E_a=\frac{1}{4\pi {ϵ}_0}\frac{q_a}{a^2}$$ $$E_b=\frac{1}{4\pi {ϵ}_0}\frac{q_b}{b^2}$$

d.

$$0$$

e. Surface charge density of conductor ${\sigma }_R$ would change, electric field outside the conductor would change.