Line charge equation in electromagnetic

[baby_1]

Hello
i have some questions about to obtain finite line charge equation.
as you know the equation of a finite line charge equation is

now i want to demonstrate this equation.
at first i assume that charge element on z>0 so
1-

i solve the problem but my final answer is different from the above answer.it seems the integral limitation give me a (-) to statement.so i change it in next step and find the correct answers
2-

as you see that answer is completely correct.

now here i have two questions
1-how can we define integral limitation?from low value to high?
2-does integral limitation change where we get charge element?

try the next steps
now i assume that charge element is z<0 so i resolve the problem
3-

as you see the answer is different from the correct answer so i assume that i get wrong intergal limitation and resolve in next step
4-

again the answer is wrong from correct.


where is my problem?
1-now how can we define integral limitation?
2-is charge element situation important?
3-why we get different answers?

any help appreciated

 

[TSny]

Hello baby_1.

The charge density ρ is positive. So, the charge dq is also positive. When you write dq = ρ dz you need to make sure dz is positive so that dq will come out positive. To make dz positive, you have to integrate in the direction of smaller to larger z. That is, integrate from –m to n. That is why your second method is correct and your first method gives the wrong overall sign.

When you are considering an element of charge dq located at negative values of z, then the $\hat{a}_z$ component of the field will be positive. But note that the expression $\frac{-zdz}{\sqrt{z^2+r^2}}\hat{a}_z$ does point in the positive z direction because z is negative. So, you would not want to write $\frac{+zdz}{\sqrt{z^2+r^2}}\hat{a}_z$ for charge elements with negative z (or positive z) because that would give the wrong direction. You should see that you can use the expression $\frac{-zdz}{\sqrt{z^2+r^2}}\hat{a}_z$ for the entire integration from –m to n.

 

[baby_1]

Thank you Tsny So much . Love you
i asking above question for two month and no one can help me .Thank you again
now if there is no problem tell me more about this questions?

1-it means if we have negative charge density ρ then dq should be negative and we should define integral limitation form +m to -n?
2- "When you are considering an element of charge dq located at negative values of z, then the a^z component of the field will be positive" why component of the field should be positive? and when we should know that the statement of integral should be modified?
3-could you introduce me a book that tell me completely about these my problems?(or a book that show all the way that you mentioned?)

Thanks

 

[TSny]

1-it means if we have negative charge density ρ then dq should be negative and we should define integral limitation form +m to -n?

No. If ρ is negative, then dq must also be negative. So, when you write dq = ρ dz, what does the sign of dz have to be? Therefore, what direction would you integrate?

2- "When you are considering an element of charge dq located at negative values of z, then the a^z component of the field will be positive" why component of the field should be positive? and when we should know that the statement of integral should be modified?

If you look at your first picture, it shows the electric field dE from a dq located at positive z (B) and a dE from a dq located at negative z (A). Which of those two electric fields is produced by the charge at negative z? Is the z component of that electric field positive or negative?

3-could you introduce me a book that tell me completely about these my problems?(or a book that show all the way that you mentioned?)

Most introductory physics texts that cover integration over charge distributions will discuss this to some extent. But I can’t think of a particular book that goes into your type of question very thoroughly. Once you think through a few examples, you will be able to handle it.

 

[baby_1]

hello dear TSny
Thank you so much for accompany me in this problem
"No. If ρ is negative, then dq must also be negative. So, when you write dq = ρ dz, what does the sign of dz have to be? Therefore, what direction would you integrate?
"
according your explanation always we have dq positive and we integrate from small size to larger size?

"Which of those two electric fields is produced by the charge at negative z? Is the z component of that electric field positive or negative?"

Thanks i understand what you say.it means we should consider the direction of electric filed to define what we should change in integral statement?

"Most introductory physics texts that cover integration over charge distributions will discuss this to some extent. But I can’t think of a particular book that goes into your type of question very thoroughly. Once you think through a few examples, you will be able to handle it"

Thanks for your suggestion.we study this line charge in "filed and wave electromagnetic by david.cheng" book and it doesn't cover more examples about line charge and integration.if you could introduce me some books or a books for more examples and integration of a line charge i would appreciate you a lot

Than you

 

[TSny]

hello dear TSny
Thank you so much for accompany me in this problem
"No. If ρ is negative, then dq must also be negative. So, when you write dq = ρ dz, what does the sign of dz have to be? Therefore, what direction would you integrate?
"
according your explanation always we have dq positive and we integrate from small size to larger size?

It's dz that would need to be positive in this situation. So, you integrate from smaller to larger z.

Thanks i understand what you say.it means we should consider the direction of electric filed to define what we should change in integral statement?

Yes, you should always draw a picture and note the direction of the field so you can get the components of the field correct (including the signs).

Thanks for your suggestion.we study this line charge in "filed and wave electromagnetic by david.cheng" book and it doesn't cover more examples about line charge and integration.if you could introduce me some books or a books for more examples and integration of a line charge i would appreciate you a lot

Most standard textbooks will have some examples. As I recall, the textbook by Paul Tipler,https://www.amazon.com/dp/1572594926/?tag=pfamazon01-20, has quite a few examples worked out in detail.