Solving for Flux Using Enclosed Charge

[InertialRef]

Homework Statement

https://www.smartphysics.com/Content/Media/Images/EM/03/h3_lineD.png

Where, λ₁ = -3.2 μC/cm, λ₂ = 9.6 μC/cm, a = 8.5 cm and h = 11.4 cm. a/2 = 4.25 cm.

What is the total flux that passes through the surface of the cylinder?

Homework Equations

E = 2kλ/r
$\Phi$ = E*A = q_enclosed/$\epsilon$_o

The Attempt at a Solution

The first thing I did was solve for the total electric field of the two line charges, which is 338823529.411765 N/C. After this, the next step would be to simply multiply by the surface area of the cylinder. However, this doesn't work because the electric field is not constant throughout the surface of the cylinder.

My next step was to try and solve it using the enclosed charge, but since I have only been given the charge densities of each line, I have no clue how to figure out enclosed charge. Is there a specific method as to how you can do this?

Any help is greatly appreciated.

 

[ehild]

Homework Statement

https://www.smartphysics.com/Content/Media/Images/EM/03/h3_lineD.png

Where, λ₁ = -3.2 μC/cm, λ₂ = 9.6 μC/cm, a = 8.5 cm and h = 11.4 cm. a/2 = 4.25 cm.

What is the total flux that passes through the surface of the cylinder?

Homework Equations

E = 2kλ/r
$\Phi$ = E*A = q_enclosed/$\epsilon$_o

The Attempt at a Solution

The first thing I did was solve for the total electric field of the two line charges, which is 338823529.411765 N/C. After this, the next step would be to simply multiply by the surface area of the cylinder. However, this doesn't work because the electric field is not constant throughout the surface of the cylinder.

My next step was to try and solve it using the enclosed charge, but since I have only been given the charge densities of each line, I have no clue how to figure out enclosed charge. Is there a specific method as to how you can do this?
Any help is greatly appreciated.

As you have noted, the electric field is not constant on the surface, so it is not AE. But the total flux is always equal to the enclosed charge divided by ε₀: you need to calculate that only from the λ-s and length h.

 

[InertialRef]

As you have noted, the electric field is not constant on the surface, so it is not AE. But the total flux is always equal to the enclosed charge divided by ε₀: you need to calculate that only from the λ-s and length h.

Alright, thank you. :) I think I got it now.

I do have one more question though. Is the method used to find the enclosed charge different depending upon what type of field you are dealing with?

 

[ehild]

I do not get you. It is valid for the relation between electric flux and enclosed charge.

ehild

 

[Nickie]

To solve for the total flux passing through the surface of the cylinder, we can use the formula \Phi = E*A, where E is the electric field and A is the surface area of the cylinder. However, as you have correctly pointed out, the electric field is not constant throughout the surface of the cylinder, so we cannot simply multiply by the surface area.

To solve this problem, we can use the concept of enclosed charge. Enclosed charge is the total charge that is enclosed within a surface. In this case, the surface is the cylinder and the enclosed charge is the total charge of both line charges that are contained within the cylinder.

To find the enclosed charge, we need to calculate the length of each line charge that is within the cylinder. This can be done by using the formula for the length of a line segment, L = \sqrt{(h-a)^2 + (a/2)^2}. In this case, L1 = 4.25 cm and L2 = 7.15 cm.

Next, we can calculate the total charge enclosed by multiplying the length of each line charge by its respective charge density. This gives us a total enclosed charge of -27.2 μC.

Now, we can use the formula \Phi = qenclosed/\epsilono to calculate the total flux passing through the surface of the cylinder. Plugging in the value for the enclosed charge and the permittivity of free space, we get a total flux of -6.8 x 10^6 Nm^2/C.

Therefore, the total flux passing through the surface of the cylinder is -6.8 x 10^6 Nm^2/C.