Biological Physics: Energy, Information, Life

232 Chapter 7. Entropic forces at work[[Student version, January 17, 2003]]

positive layer

negative surface

E(x 1 )

E(x 2 )

−σqdA

dA

ρq(x)dAdx

dA

dx

x

Figure 7.7:(Schematic.) A planar distribution of charges. A thin sheet of negative charge (hatched, bottom)lies
next to a neutralizing positive layer of free counterions (shaded, top). The individual counterions are not shown;
the shading represents their average density. The lower box encloses a piece of the surface, and so contains total
charge−σqdA,where dAis its cross-sectional area and−σqis the surface charge density. The upper box encloses
chargeρq(x)dAdx,whereρq(x)isthe charge density of counterions. The electric fieldE(x)atanypoint equals the
electrostatic force on a small test particle at that point, divided by the particle’s charge. For all positivex,the field
points along the−ˆxdirection. The field atx 1 is weaker than that atx 2 ,because of the repelling layer of positive
charge betweenx 1 andx=0is thicker than that betweenx 2 andx=0.

sheet with uniform surface charge density−σq,next to a spread-out layer of positive charge with
volume charge densityρq(x). Thusσqis a positive constant with unitscoul m−^2 ,whileρq(x)isa
positive function with unitscoul m−^3 .Everything is constant in theyˆandˆzdirections.
The electric field above the negative sheet is a vector pointing along the−ˆxdirection, so its
x-component isEx≡E,where the functionE(x)iseverywhere negative. Just above the sheet the
electric field is proportional to the surface charge density:

E|surface=−σq/ε. Gauss Lawat a flat, charged surface (7.18)

In this formula, the permittivityεis the same constant appearing in Your Turn 7d; in water it’s
about 80 times the value in air or vacuum.^5 As we move away from the surface, the field gets
weaker (less negative): A positively charged particle is still attracted to the negative layer, but the
attraction is partially offset by the repulsion of the intervening positive charges. Thedifferencein
electric fields at two nearby points reflects the reduction of Equation 7.18 by the charge per area
in the space between the points. Calling the pointsx±^12 dx,this surface charge density equals
ρq(x)dx(see Figure 7.7). Hence

E(x+^12 dx)−E(x−^12 dx)=(dx)ρq(x)/ε. (7.19)

In other words,
dE
dx

=

ρq

ε

. Gauss Lawin bulk (7.20)

Section 7.4.3 will use this relation to find the electric field everywhere outside the surface.
T 2 Section 7.4.2′on page 250 relates the discussion above to the more general form of the Gauss
Law.

(^5) Many authors use the notation 0 for the quantity calledεin this book. It’s a confusing notation, since then
their≈80 is dimensionless while 0 (which equals ourε 0 )doeshave dimensions.