Integrating electric field of rod

In summary: There are several ways to proceed when trying to integrate the y-component of the electric field of a rod.
  • #1
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Homework Statement
Please see below
Relevant Equations
Please see below
For this problem,
1674105439521.png

However, I am trying to solve this problem using an alternative method compared with the solutions. My method is:

##\vec E = k_e \int \frac {dq} {r^2} \, dx ## ##\hat r##
##\vec E = k_e \int \frac {\lambda} {x^2 + d^2} \, dx## ## \hat r##

If I let ## \hat r = \frac {-x\hat i + d\hat j} {\sqrt {x^2 + d^2}}## then I get the same answer as the solutions.

However, how dose ## \hat r = \frac {-x\hat i + d\hat j} {\sqrt {x^2 + d^2}}##?

I see intuitively that the ##\hat r## dose point in the same direction as the ##d\vec E## for each charge segment ##dq##.

Many thanks!
 
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  • #2
Callumnc1 said:
##\vec E = k_e \int \frac {dq} {r^2} \, dx ## ##\hat r##
You mean ##\vec E = k_e \int \frac {dq} {r^2} \hat r##
Callumnc1 said:
dose ## \hat r = \frac {-x\hat i + d\hat j} {\sqrt {x^2 + d^2}}##?
Of course. What is the vector from (x,0) to (0,d)?
What is its magnitude?
So what is the unit vector pointing from (x,0) to (0,d)?
 
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  • #3
haruspex said:
You mean ##\vec E = k_e \int \frac {dq} {r^2} \hat r##

Of course. What is the vector from (x,0) to (0,d)?
What is its magnitude?
So what is the unit vector pointing from (x,0) to (0,d)?
Thank you for your reply @haruspex ! Yes, thanks for pointing that out: I did mean ##\vec E = k_e \int \frac {dq} {r^2} \hat r##.

The vector pointing from (x,0) to (0,d) is ##\vec r = -x\hat i + d\hat j ## and its magnitude is ##\sqrt{x^2 + d^2}## so I guess we must divide the ##\vec r## but it's magnitude to find its direction to get ## \hat r = \frac {-x\hat i + d\hat j} {\sqrt {x^2 + d^2}}##

Then we just translate the unit vector ##\hat r## to the right for a charge element ##dq## as shown below:
1674120168371.png

Is my reasoning correct please?

Many thanks!
 

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  • #4
Callumnc1 said:
Thank you for your reply @haruspex ! Yes, thanks for pointing that out: I did mean ##\vec E = k_e \int \frac {dq} {r^2} \hat r##.

The vector pointing from (x,0) to (0,d) is ##\vec r = -x\hat i + d\hat j ## and its magnitude is ##\sqrt{x^2 + d^2}## so I guess we must divide the ##\vec r## but it's magnitude to find its direction to get ## \hat r = \frac {-x\hat i + d\hat j} {\sqrt {x^2 + d^2}}##

Then we just translate the unit vector ##\hat r## to the right for a charge element ##dq## as shown below:View attachment 320665
Is my reasoning correct please?

Many thanks!
Yes.
 
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  • #5
haruspex said:
Yes.
Ok thank you for your help @haruspex !
 
  • #6
Thank you for you reply on my other thread @haruspex! That thread has been taken down as I did not realize that I was not allowed to post another thread about another question I had about this problem. In the future I will try to get out all my questions about the same problem in one thread.

So my question was about integrating the y-component of the electric field of the rod. I am not sure how to integrate ## E_y = k_ed\lambda\int \frac {dx} {(x^2 + d^2)^{3/2}} \ ##

Many thanks!
 
  • #7
Callumnc1 said:
Thank you for you reply on my other thread @haruspex! That thread has been taken down as I did not realize that I was not allowed to post another thread about another question I had about this problem. In the future I will try to get out all my questions about the same problem in one thread.

So my question was about integrating the y-component of the electric field of the rod. I am not sure how to integrate ## E_y = k_ed\lambda\int \frac {dx} {(x^2 + d^2)^{3/2}} \ ##

Many thanks!
There are several ways to proceed.
You can try using a trig substitution to get rid of the square root.
Another way is to turn x.dx into d(x2)/2.
Another is to guess that ##(x^2 + d^2)^{1/2}## may appear in the answer, and differentiate that to see what you get.

Sorry… just realised you are asking about ## E_y = k_ed\lambda\int \frac {dx} {(x^2 + d^2)^{3/2}} \ ##, not ## E_y = k_ed\lambda\int \frac {xdx} {(x^2 + d^2)^{3/2}} \ ##, so the last two won't work.

Bear in mind that (1) an answer in the form of an integral may be acceptable for part a), and (2) part b) asks for an approximation, which suggests that one or both of the integrals cannot be solved exactly.
 
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  • #8
haruspex said:
There are several ways to proceed.
You can try using a trig substitution to get rid of the square root.
Another way is to turn x.dx into d(x2)/2.
Another is to guess that ##(x^2 + d^2)^{1/2}## may appear in the answer, and differentiate that to see what you get.

Sorry… just realised you are asking about ## E_y = k_ed\lambda\int \frac {dx} {(x^2 + d^2)^{3/2}} \ ##, not ## E_y = k_ed\lambda\int \frac {xdx} {(x^2 + d^2)^{3/2}} \ ##, so the last two won't work.

Bear in mind that (1) an answer in the form of an integral may be acceptable for part a), and (2) part b) asks for an approximation, which suggests that one or both of the integrals cannot be solved exactly.
Thank you for your reply @haruspex !

I can solve the first integral for the x-component of the electric field, however, I have not done trig substitution yet.

All the solutions say for the integration of the y-component of the electric field is:
1674147960714.png

Thank you!
 
  • #9
Callumnc1 said:
Thank you for your reply @haruspex !

I can solve the first integral for the x-component of the electric field, however, I have not done trig substitution yet.

All the solutions say for the integration of the y-component of the electric field is:
View attachment 320704
Thank you!
You can check that easily enough by differentiating the result.
For trig substitution, what two trig functions do you know that satisfy ##g^2=f^2+1##?
 
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  • #10
haruspex said:
You can check that easily enough by differentiating the result.
For trig substitution, what two trig functions do you know that satisfy ##g^2=f^2+1##?
Thank you for your reply @haruspex!

There are two trig identities I'm aware of that satisfy ##g^2=f^2+1##:

1. ## \sec^2\theta = \tan^2\theta + 1##
2. ## \csc^2\theta = \cot^2\theta +1 ##

Which two trig functions were you looking for?

Thank you!
 
  • #11
Callumnc1 said:
Thank you for your reply @haruspex!

There are two trig identities I'm aware of that satisfy ##g^2=f^2+1##:

1. ## \sec^2\theta = \tan^2\theta + 1##
2. ## \csc^2\theta = \cot^2\theta +1 ##

Which two trig functions were you looking for?

Thank you!
The first will do. Can you see how to use it to get rid of the square root in the denominator?
 
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  • #12
haruspex said:
The first will do. Can you see how to use it to get rid of the square root in the denominator?
Thanks for your reply @haruspex ! I think so - I'll give it I try!
 
  • #13
haruspex said:
The first will do. Can you see how to use it to get rid of the square root in the denominator?
So far I have got:

## (\sec^2\theta)^{1/3} = (\tan^2\theta + 1)^{1/3}##
## (\sec\theta)^{2/3} = (\tan^2\theta + 1)^{1/3}##
## \sec\theta = d^2 + x^2 ## from comparison
## x = (\sec\theta - d^2)^{1/2} ##
## dx = \frac {\sec\theta\tan\theta} {2\sqrt{\sec\theta - d^2}} d\theta ##
 
  • #14
Callumnc1 said:
## \sec\theta = d^2 + x^2 ##
No, you can't do that. It's dimensionally invalid.
Besides, you want ##d^2 + x^2 ## to be something squared so that you can take its square root.
Try ##x=d\times## a trig function.
 
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  • #15
Callumnc1 said:
So far I have got:

## (\sec^2\theta)^{1/3} = (\tan^2\theta + 1)^{1/3}##
## (\sec\theta)^{2/3} = (\tan^2\theta + 1)^{1/3}##
## \sec\theta = d^2 + x^2 ## from comparison
## x = (\sec\theta - d^2)^{1/2} ##
## dx = \frac {\sec\theta\tan\theta} {2\sqrt{\sec\theta - d^2}} d\theta ##
Thanks for using LaTeX !

You have some things mixed up. Also, using "d" (lower case D) as a symbol in this problem is a bad idea, but that's what was given to you., so let's go with it.

Actually, you want to let ##\displaystyle \tan \theta = \dfrac{x}{(d)} ## , so that ##\displaystyle (d) \tan \theta = x ##

That gives you ##\displaystyle ((d)^2 + x^2)^{(3/2)}=((d)^2 + (d)^2 \tan^2 \theta)^{(3/2)}=(d)^3(1 +\tan^2 \theta)^{(3/2)}## for your denominator.

The change of variable gives you ##\displaystyle dx = (d)\sec^2 \theta \, d\theta## .
 
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  • #16
SammyS said:
Thanks for using LaTeX !

You have some things mixed up. Also, using "d" (lower case D) as a symbol in this problem is a bad idea, but that's what was given to you., so let's go with it.

Actually, you want to let ##\displaystyle \tan \theta = \dfrac{x}{(d)} ## , so that ##\displaystyle (d) \tan \theta = x ##

That gives you ##\displaystyle ((d)^2 + x^2)^{(3/2)}=((d)^2 + (d)^2 \tan^2 \theta)^{(3/2)}=(d)^3(1 +\tan^2 \theta)^{(3/2)}## for your denominator.

The change of variable gives you ##\displaystyle dx = (d)\sec^2 \theta \, d\theta## .
Sorry for my very late reply - Thank you @haruspex and @SammyS for your replies! I'll see if I can try that out tomorrow, sorry I'm a bit busy at the moment.
haruspex said:
No, you can't do that. It's dimensionally invalid.
Besides, you want ##d^2 + x^2 ## to be something squared so that you can take its square root.
Try ##x=d\times## a trig function.
Also, was it it dimensionally invalid? Is it because there is meters (which dimension is length) on one side and no dimensions on the other side?

Thank you!
 
  • #17
Callumnc1 said:
Is it because there is meters (which dimension is length) on one side and no dimensions on the other side?
Yes.
 
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  • #18
haruspex said:
Yes.
Thank you @haruspex ! I don't see how you get a conflicting result when you change the units, to cm say. Could you please explain?

Thank you!
 
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  • #19
SammyS said:
Thanks for using LaTeX !

You have some things mixed up. Also, using "d" (lower case D) as a symbol in this problem is a bad idea, but that's what was given to you., so let's go with it.

Actually, you want to let ##\displaystyle \tan \theta = \dfrac{x}{(d)} ## , so that ##\displaystyle (d) \tan \theta = x ##

That gives you ##\displaystyle ((d)^2 + x^2)^{(3/2)}=((d)^2 + (d)^2 \tan^2 \theta)^{(3/2)}=(d)^3(1 +\tan^2 \theta)^{(3/2)}## for your denominator.

The change of variable gives you ##\displaystyle dx = (d)\sec^2 \theta \, d\theta## .
No worries @SammyS ! So solving using trig substitution:

## E_y = k_ed\lambda\int \frac {dx} {(x^2 + d^2)^{3/2}} \ ##
## E_y = k_ed\lambda\int \frac {(d)\sec^2 \theta \, d\theta} {(d)^3(1 +\tan^2 \theta)^{(3/2)}} \ ##
## E_y = k_ed\lambda\int \frac {\sec^2 \theta \, \theta} {(d)^2(1 +\tan^2 \theta)^{(3/2)}} \ ##
## E_y = k_ed\lambda\int \frac {1 +\tan^2 \theta} {(d)^2(1 +\tan^2 \theta)^{(3/2)}} \ ##
## E_y = k_ed\lambda\int \frac {1} {(d)^2(1 +\tan^2 \theta)^{(1/2)}} \ ##

I'm not sure how to finish from here. Have I done it correctly so far?

EDITED II: I also found this on an integral table:
1674539966760.png

However, I think this only holds for when the power of the brackets for the bottom fraction is equal to 1. Am I correct?

Many thanks!
 
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  • #20
Callumnc1 said:
Thank you @haruspex ! I don't see how you get a conflicting result when you change the units, to cm say. Could you please explain?

Thank you!
Suppose d and x are 3cm and 4cm. Your expression in post #13 gives ##\sec(\theta)=25##. But putting them in m gives ##\sec(\theta)=.00025##.
 
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  • #21
Callumnc1 said:
how to finish from here.
Instead of converting ##\sec^2## to ##1+\tan^2## in the numerator, try the reverse conversion in the denominator.
Btw, you have dropped ##d\theta##.
 
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  • #22
Callumnc1 said:
No worries @SammyS ! So solving using trig substitution:

## E_y = k_ed\lambda\int \frac {dx} {(x^2 + d^2)^{3/2}} \ ##
## E_y = k_ed\lambda\int \frac {(d)\sec^2 \theta \, d\theta} {(d)^3(1 +\tan^2 \theta)^{(3/2)}} \ ##
## E_y = k_ed\lambda\int \frac {\sec^2 \theta \, \theta} {(d)^2(1 +\tan^2 \theta)^{(3/2)}} \ ##
## E_y = k_ed\lambda\int \frac {1 +\tan^2 \theta} {(d)^2(1 +\tan^2 \theta)^{(3/2)}} \ ##
## E_y = k_ed\lambda\int \frac {1} {(d)^2(1 +\tan^2 \theta)^{(1/2)}} \ ##

I'm not sure how to finish from here. Have I done it correctly so far?

EDITED: I also found this on an integral table:
View attachment 321000
However, I think this only holds for when the power of the brackets for the bottom fraction is equal to 1.
Change ##\displaystyle 1+ \tan^2 \theta## to ##\displaystyle \sec^2 \theta## rather than the other way around. Pull the ##\dfrac 1 {(d)^2}## out of the integral. Also, you have somehow dropped the ##d\theta## from the integral. (maybe a typo)
 
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  • #23
haruspex said:
Suppose d and x are 3cm and 4cm. Your expression in post #13 gives ##\sec(\theta)=25##. But putting them in m gives ##\sec(\theta)=.00025##.
Thank you @haruspex for the example - I see now!
 
  • #24
Thank you for your replies @SammyS and @haruspex ! Sorry it dose look like I dropped the ##d\theta##

## E_y = k_ed\lambda\int \frac {\sec^2 \theta \, d\theta} {(d)^2(sec^2 \theta)^{(3/2)}} \ ##
## E_y = k_ed^{-1}\lambda\int \frac {\sec^2 \theta \, d\theta} {(sec \theta)^3} \ ##
## E_y = k_ed^{-1}\lambda\int \frac {1} {(sec \theta)} \, d\theta##
## E_y = k_ed^{-1}\lambda\int cos\theta \, d\theta##
## E_y = k_ed^{-1}\lambda sin\theta ##

Is it so far correct? I should probably add some limits of integration e.g ## \theta_1 = 0 ## and ## \theta_2 = 90 ##
 
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  • #25
Callumnc1 said:
Thank you for your replies @SammyS and @haruspex ! Sorry it dose look like I dropped the ##d\theta##

## E_y = k_ed\lambda\int \frac {\sec^2 \theta \, d\theta} {(d)^2(sec^2 \theta)^{(3/2)}} \ ##
## E_y = k_ed^{-1}\lambda\int \frac {\sec^2 \theta \, d\theta} {(\sec \theta)^3} \ ##
## E_y = k_ed^{-1}\lambda\int \frac {1} {(\sec \theta)} \, d\theta##
## E_y = k_ed^{-1}\lambda\int \cos\theta \, d\theta##
## E_y = k_ed^{-1}\lambda \sin\theta ##

Is it so far correct? I should probably add some limits of integration e.g ## \theta_1 = 0 ## and ## \theta_2 = 90 ##
Not the correct limits of integration.
You can write ##\sin \theta## in terms of ##x## and integrate from ##0## to ##L##, the original limits of integration.
Or you can find the limits of integration you need for ##\theta##.
 
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  • #26
SammyS said:
Not the correct limits of integration.
You can write ##\sin \theta## in terms of ##x## and integrate from ##0## to ##L##, the original limits of integration.
Or you can find the limits of integration you need for ##\theta##.
Thank you for you reply @SammyS !

## E_y = k_ed^{-1}\lambda\int_0^{\theta_L} cos\theta \, d\theta##
## E_y = k_ed^{-1}\lambda sin\theta_L ##

Where ##\theta_L## is the angle the position vector makes with in the ##-\hat j## direction to the end of the rod.
1674594221333.png


## E_y = k_e\lambda \frac {L} {d(L^2 + d^2)^{1/2}} ##
## E_y = \frac {k_eQ} {d(L^2 + d^2)^{1/2}} ##

Which is the same answer as the solutions! :)
 
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FAQ: Integrating electric field of rod

What is the general approach to integrating the electric field of a uniformly charged rod?

The general approach involves dividing the rod into infinitesimal charge elements, calculating the electric field produced by each element at a point of interest, and then integrating these contributions over the entire length of the rod. This typically requires setting up an integral with respect to the position along the rod and using Coulomb's law for the electric field produced by a point charge.

How do you set up the integral for the electric field of a rod along its axis?

To set up the integral for the electric field along the axis of a uniformly charged rod, you first define the linear charge density (λ), which is the total charge (Q) divided by the length (L) of the rod. Then, you consider a small charge element (dq = λ dx) at a position x along the rod. The electric field at a point on the axis is found by integrating the contributions of all these elements, taking into account the distance from each element to the point of interest.

What are the boundary conditions for integrating the electric field of a finite rod?

The boundary conditions depend on the position of the point where you want to calculate the electric field. For a rod of length L, if the point is located at a distance d from one end of the rod, the limits of integration will be from 0 to L for the position along the rod. The distance to the point of interest will vary from (d) to (d + L) depending on the position of the charge element along the rod.

How do you handle the vector nature of the electric field when integrating for a rod not along its axis?

When the point of interest is not along the axis of the rod, the electric field contributions from each infinitesimal charge element have both x and y components (or radial and tangential components in cylindrical coordinates). You need to resolve the electric field into these components, integrate each component separately, and then combine the results vectorially. Symmetry can often be used to simplify the problem, such as noting that certain components may cancel out.

Can you provide an example of integrating the electric field of a rod along its perpendicular bisector?

For a rod of length L with uniform charge density λ, the electric field at a point along the perpendicular bisector at a distance d from the center involves integrating the contributions from each infinitesimal charge element. The symmetry of the problem means that only the perpendicular components of the electric field need to be considered. The integral is set up by considering an element at position x from the center, with limits from -L/2 to L/2. The distance to the point is sqrt(x^2 + d^2), and the electric field component is integrated

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