Uniform Slab-Finding Electric Field Using Gauss Law

[Arman777]

Homework Statement

Uniform Slab: Consider an infinite slab of charge with thickness 2a. We choose the origin inside the slab at an equal distance from both faces (so that the faces of the slab are at z = +a and z = −a). The charge density ρ inside the slab is uniform (i.e., ρ =const). Consider a point with coordinates (x,y,z). Using Gauss’ law, find the electric field
(a) when the point is inside the slab (−a < z < +a),
(b) and when the point is outside the slab (z > a or z < −a).
(c) Sketch the Ez vs z graph.
(d) If the density was not constant at its a function of z like $ρ=Bz^2$ then calculate the upper steps again.

Homework Equations

Gauss Law

The Attempt at a Solution

a) I took a cylinder Gaussian surface inside the slab forand from that I found $E=\frac {ρz} {2ε_0}$ .z is the height of the point that we choose from the origin.
b)I took a cylinder again and from that I found $E=\frac {ρa} {2ε_0}$
c)The field will be constant cause ρ and a is constant also $ε_0$ so As z increases it inrease until a.And from that its constant.
d)Then Electric field will be $E=\frac {Bz^3} {6ε_0}$ for inside , $E=\frac {Ba^3} {6ε_0}$ for outside ?
Is these true ?

 

[TSny]

The Attempt at a Solution

a) I took a cylinder Gaussian surface inside the slab forand from that I found $E=\frac {ρz} {2ε_0}$ .z is the height of the point that we choose from the origin.

You have the right approach. But your answer is off by a numerical factor. Can you show in more detail how you got your result? In particular, how was your cylindrical Gaussian surface oriented and positioned?

 

[Arman777]

I took a cylinder like height is z and radius is x.So $Eπx^2=Q/ε_0$ $Q=ρπx^2z$ But there's two sides so we should multiply this by 2 so $2Eπx^2=\frac {Q} {ε_0}$ and then $E=\frac {pz} {2ε_0}$

 

[Arman777]

Oh wait

 

[Arman777]

Height is $2z$

 

[Arman777]

$E=\frac {ρz} {ε_0}$ ?

 

[TSny]

$E=\frac {ρz} {ε_0}$ ?

Yes.

 

[Arman777]

Then (b) is $E=\frac {pa} {ε_0}$
(c) will be the same (it increase until $z=a$ then its constant)
for (d) inside $E=\frac {Bz^3} {3ε_0}$ and outside $E=\frac {Ba^3} {3ε_0}$ and the graph again rises at z rises then its constant ??

 

[TSny]

Looks good. Make sure your graphs include negative values of z as well as positive values of z. So you will need to think about the sign of E_z for negative z.

 

[Arman777]

I see thanks