Lecture 15

Readings

http://umdberg.pbworks.com/w/page/50987852/Screening%20of%20electrical%20interactions%20in%20salt%20solution%20%28draft%29

http://umdberg.pbworks.com/w/page/51186725/Debye%20length

Conductors , Process

• Process

  • In metal, many electrons are available.

    • They are free to move

  • If you put a charge inside a metal, or near a metal, these free electrons rearrange so as to completely cancel any electric field inside the metal:

• • charges in metal are basically "absence of electrons" at that location

Debye Length

• In salt solution, there are quite a few free charges, in the form of ions now.

  • Both + and - charges now

    • Not quite as many as in a conductor
    • These ions are affected by thermal motion.

• If you put a charged object into a salt solutions, ions form a "cloud" around the object, which partially cancels the electric field that the charged object would have caused.

  • In a metal, the electric field was perfectly canceled

• Without a salt solution, say in a vacuum, the electric field gets weaker as $\frac{1}{r^2}$$\frac{1}{r^2}$

• With this cloud present in a salt solution, it will get weaker even faster as $r$$r$ increases.

• There is a quantity called the Debye Length which describes how fast (at what distance) the electric field gets weaker

  • The derivation is complicated

  • Essentially it's a balance between potential energy and thermal energy

  • Potential energy is due to:

    • positive charge in above picture
    • Counter-ions (ions with opposite charge compared to object) being attracted to the object; and
    • Counter-ions being repelled by each other when they all try to crowd around object

  • Thermal motion/entropy wants to keep all the ions evenly spread out.

• The result is that the electric field ( and electric potential also ) fall off according to:

• E field falls off a little faster than this

• This equation indicates that if the distance from the charged object is $r={\lambda }_D$$r = \lambda_D$ , then $V$$V$ and $E$$E$ are about 0.37 as strong as they are when right next to the object

• Compare this to vaccum ( no salt solution ).

  • There, electric potential $V$$V$ falls off as $\frac{1}{r}$$\frac{1}{r}$ , which means $V$$V$ extends much farther in vacuum
  • ${\lambda }_D$$\lambda_D$ is essentially "the distance the E field penetrates into the solutions"

• Where z = charge per ion and c = concentration of ions
• As $T$$T$ increases , ${\lambda }_D$$\lambda_D$ increases
• As $z$$z$ increases, ${\lambda }_D$$\lambda_D$ decreases
• As $c$$c$ increases, ${\lambda }_D$$\lambda_D$ decreases
• Another way of thinking of this is that the charges on an object ( or more commonly charges on some molecule ) are "screened" in a salt solutions
• The effects of the charged object are "partially covered up" or "attenuated"