What is the Charge Density in the Region 0<z<1 Using Gauss's Law?

[Jahnavi]

Homework Statement

Homework Equations

The Attempt at a Solution

In the region |z|<1 ,

E(z) = -dV/dz = 10zk

This means there is a variable electric field in the region -1<z<1 .In the +z region it is directed in the +z direction and vica versa .

For finding the charge density in the region 0<z<1, I need to apply Gauss's law . I am not supposed to use differential form of Gauss law .

Consider at a distance z from the origin a thin cuboid of side a×a and height ∆z .

The two square faces are at distances z₁ and z₂ , z₂>z₁

Flux through the closed cuboidal box would have a contribution from only these two square surfaces which are perpendicular to the electric field .

Net flux = E(at z₂)a² - E(at z₁)a² = 10a²z₂-10a²z₁

Applying Gauss law ∫E⋅ds= ρa²∆z/ε

10a²z₂-10a²z₁ = ρa²∆z/ε

10a²∆z = ρa²∆z/ε

ρ = 10ε in the region 0<z<1

Is that right ?

Please help me with this problem .

 

[TSny]

Your work looks correct to me.

 

[Jahnavi]

Thank you very much .

Sorry , I had constantly been editing my work till I saw your reply (after posting the thread ) to make things clearer. I hope I haven't changed anything which you found correct and which is wrong now .

The electric field is directed in the +z direction for 0<z<1 .

For z>1 , there is a constant electric field in the +z direction .

Here I need to prove that there is no charge anywhere in z>1 .

For this I can again consider a cubical shaped box of dimensions a×a×a , such that there will be two square surfaces perpendicular to the electric field .

Since the electric field is constant the net flux through the closed cubical surface would be zero .

Applying Gauss law the net charge enclosed would be zero .

This means there would be no charge anywhere in the region z>1 .

Is that sufficient reasoning ?

Would 1) be the correct option ?

 

[TSny]

That looks good. Strictly speaking, I think your arguments yield the average charge density, ρ_avg, in the rectangular boxes you used. But it should be easy to go on to argue how to get the charge density, ρ, at any point.

 

[Jahnavi]

But it should be easy to go on to argue how to get the charge density, ρ, at any point.

If we consider a cubical box of dimensions of infinitesimal length dx , dy , dz , do we get charge density at a point .Is this what you are indicating at ?

 

[TSny]

If we consider a cubical box of dimensions of infinitesimal length dx , dy , dz , do we get charge density at a point .Is this what you are indicating at ?

Yes. Good work.

 

[Jahnavi]

Thanks !