Calculating Electric Field Using Gauss' Law for a Charged Insulating Slab

[Elvis 123456789]

Homework Statement

A slab of insulating material has thickness 2d and is oriented so that its faces are parallel to the yz-plane and given by the planes x=d and x=−d. The y- and z-dimensions of the slab are very large compared to d and may be treated as essentially infinite. Let the charge density of the slab be given by ρ(x)=ρ0(x/d)^2 where ρ0 is a positive constant.Using Gauss's law, find the magnitude of the electric field due to the slab at 0<|x|<d.

Homework Equations

ρ=Q/V

=∫E*dA= (Q_encl)/ϵ0

The Attempt at a Solution

I started by choosing a cylinder as my gaussian surface; I placed one face of the cylinder on X=0(parallel with the yz plane) and the other on X=x <d. By symmetry, the electric flux simplifies to EA=(Q_encl)/ϵ0

=> EA=(ρ(x)V_encl)/ϵ0
=>EA=(ρ(x)Ax)/ϵ0
=> E=(ρ(x)*x)/ϵ0
=>E=(ρ0*x^3)/(ϵ0*d^2)
this is wrong and I don't know why. The correct answer contains a 3 in the denominator but I don't know why

 

[SammyS]

Homework Statement

A slab of insulating material has thickness 2d and is oriented so that its faces are parallel to the yz-plane and given by the planes x=d and x=−d. The y- and z-dimensions of the slab are very large compared to d and may be treated as essentially infinite. Let the charge density of the slab be given by ρ(x)=ρ0(x/d)2 where ρ0 is a positive constant.Using Gauss's law, find the magnitude of the electric field due to the slab at 0<|x|<d.

Homework Equations

ρ=Q/V

=∫E*dA= (Q_encl)/ϵ0

The Attempt at a Solution

I started by choosing a cylinder as my gaussian surface; I placed one face of the cylinder on X=0(parallel with the yz plane) and the other on X=x <d. By symmetry, the electric flux simplifies to EA=(Q_encl)/ϵ0

=> EA=(ρ(x)V_encl)/ϵ0
=>EA=(ρ(x)Ax)/ϵ0
=> E=(ρ(x)*x)/ϵ0
=>E=(ρ0*x^3)/(ϵ0*d^2)
this is wrong and I don't know why. The correct answer contains a 3 in the denominator but I don't know why

Did you mean that the charge density is ρ(x)=ρ₀(x/d)2 or is that 2 an exponent?

I suspect that you meant ρ(x)=ρ₀(x/d)²

The three in the denominator is the result of taking the anti-derivative of x².

 

[Suraj M]

Could you please edit the equation for charge density? The "2" doesn't look like its a super script
Firstly, in the last four steps , the A on the LHS is not the same A on the RHS, try drawing it,
Secondly, do you think you can just substitute for ##\rho (x)## as given in the question
The charge density varies with radius of the Gaussian surface you take
First start off by calculating total charge enclosed.

 

[Elvis 123456789]

Could you please edit the equation for charge density? The "2" doesn't look like its a super script
Firstly, in the last four steps , the A on the LHS is not the same A on the RHS, try drawing it,
Secondly, do you think you can just substitute for ##\rho (x)## as given in the question
The charge density varies with radius of the Gaussian surface you take
First start off by calculating total charge enclosed.

I can see now how just substituting for ρ(x) would be incorrect, but I don' see why the areas on both sides aren't the same. The area on the left hand side is the area=pi*r^2 of the face on the cylinder, and on the right hand side, the volume is the same area=pi*r^2 multiplied by the length of the cylinder "x"

 

[Elvis 123456789]

Did you mean that the charge density is ρ(x)=ρ₀(x/d)2 or is that 2 an exponent?

I suspect that you meant ρ(x)=ρ₀(x/d)²

The three in the denominator is the result of taking the anti-derivative of x².

Yes I typed it in incorrectly. So I must first integrate the charge density to find the total charge enclosed by the gaussian surface?

 

[SammyS]

Yes I typed it in incorrectly. So I must first integrate the charge density to find the total charge enclosed by the gaussian surface?

Yes, you must integrate.

(The areas are equal on both sides. You were right about that. )

 

[Elvis 123456789]

Yes, you must integrate.

(The areas are equal on both sides. You were right about that. )

ok so I first imagine that the gaussian surface has an infinitesimal length "dx" which encloses an infinitesimal charge "dQ".

then dQ=ρ(x)*dV
=> dQ=ρ(x)*A*dx
=> dQ=ρ0(x/d)^2 *A*dx
=>Q_enlcosed=(ρ0*A)/(d^2)∫(x^2)dx (integral from X=0 to X=x < d)
=> Q_enclosed= (ρ0*A*x^3)/(3*d^2)
then plugging into gauss' law

=> EA= (ρ0*A*x^3)/(3*d^2)*1/ϵ0
=> E=(ρ0*x^3)/(3*ϵ0*d^2)
is it correct to think about it this way?

 

[Suraj M]

(The areas are equal on both sides. You were right about that. )

Hi sammy
I don't mean to confuse the OP, so may I ask how the areas are the same I always thought that the areas in EdA is the curved surface of the cylinder and the area on the RHS( in this case) would be a circle in the XY plane with the length (which is infinity) along z axis

 

[Suraj M]

Hi sammy
I don't mean to confuse the OP, so may I ask how the areas are the same I always thought that the areas in EdA is the curved surface of the cylinder and the area on the RHS( in this case) would be a circle in the XY plane with the length (which is infinity) along z axis

Oh okay, I took the wrong cylinder, even then the area on the LHS is it just the circular surface parallel to the YZ plane ?

 

[Elvis 123456789]

Oh okay, I took the wrong cylinder, even then the area on the LHS is it just the circular surface parallel to the YZ plane ?

While we're on the topic, I'll add why I think they're the same so anybody can correct me if I'm wrong.
The area in the LHS of gauss' law(∫E*dA) represents the area of the closed surface that contributes to the electric flux; in other words, if some area of the gaussian surface doesn't have a perpendicular component of electric field going through it, then it doesn't show up on the left hand side. In the problem I posted, only the face of the cylinder at X=x < d, has some electric field going through it; the area of the curved surface doesn't contribute because everywhere along the curved surface the electric field is parallel to the surface. On the RHS the other area comes from the total enclosed charge (Q_enclosed) which can usually be substituted for ρ*V_enclosed (if the charge density is uniform as I have found out in this problem). Then the enclosed volume(V_enclosed) would be the area of the face on the cylinder(A=pi*r^2) multiplied by the length of the cylinder(x).

 

[SammyS]

Oh okay, I took the wrong cylinder, even then the area on the LHS is it just the circular surface parallel to the YZ plane ?

Right !

 

[Suraj M]

Thanks Elvis for posting
I learned a lot from it too [emoji2]

 

[Elvis 123456789]

Thanks Elvis for posting
I learned a lot from it too [emoji2]

Thank you for helping!