Field of a Charged Rod: Difference between revisions
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''' | '''Milo Karnes, Spring 2025''' | ||
== The Main Idea == | == The Main Idea == | ||
In earlier studies, we learned about the electric field created by a point charge. However, in the real world, charges are often spread out over objects with shape and structure. One common example is a '''uniformly charged rod'''. To determine the electric field from such an object, we divide the rod into many infinitesimally small charge segments, treat each as a point charge, and integrate their contributions. | |||
The key idea is to approximate the rod as a continuous line of charge using the principle of superposition. We consider the symmetry of the setup to simplify the problem and focus on the components of the electric field that don't cancel out. | |||
The process of finding the electric field from a charged rod involves four main steps: | |||
# '''Model the rod as many small charge elements''' and draw the electric field vector <math>\Delta \vec{E}</math> from a single element. | |||
# '''Use symmetry''' to argue which components cancel and which remain. | |||
# '''Integrate''' the contributions from all elements to find the net electric field. | |||
# '''Verify''' that the result makes physical sense (units, direction, and limiting behavior). | |||
== A Mathematical Model == | |||
To calculate the electric field of a uniformly charged rod, we treat the rod as a continuous distribution of charge. Let the rod have total length <math>L</math> and total charge <math>Q</math>, centered along the x-axis. The observation point is located on the y-axis a distance <math>y</math> above the center. | |||
=== Step 1: Break the Rod into Pieces === | |||
We divide the rod into tiny segments of length <math>dx</math>. Each segment behaves like a point charge: | |||
<math> dq = \lambda \, dx </math>, where <math> \lambda = \frac{Q}{L} </math> is the linear charge density. | |||
Each <math>dq</math> contributes a small electric field <math> d\vec{E} </math> at the observation point. | |||
[[Image:ChargedRodBreakdown.png|600px|center|thumb|Dividing the rod into segments, each producing a small electric field]] | |||
=== Step 2: Write the Field Expression for One Piece === | |||
Using Coulomb’s Law, the electric field contribution from one element is: | |||
<math> | |||
d\vec{E} = \frac{1}{4\pi\varepsilon_0} \cdot \frac{dq}{r^2} \cdot \hat{r} | |||
</math> | |||
The vector from the source to the observation point is: | |||
<math> \vec{r} = \langle 0, y \rangle - \langle x, 0 \rangle = \langle -x, y \rangle </math> | |||
So: | |||
<math> r = \sqrt{x^2 + y^2} \quad \text{and} \quad \hat{r} = \frac{\langle -x, y \rangle}{\sqrt{x^2 + y^2}} </math> | |||
\ | |||
Putting it all together: | |||
<math> | |||
d\vec{E} = \frac{1}{4\pi\varepsilon_0} \cdot \frac{\lambda \, dx}{(x^2 + y^2)} \cdot \frac{\langle -x, y \rangle}{\sqrt{x^2 + y^2}} | |||
= \frac{\lambda}{4\pi\varepsilon_0} \cdot \frac{\langle -x, y \rangle \, dx}{(x^2 + y^2)^{3/2}} | |||
</math> | |||
Note: The x-components of the field cancel due to symmetry. Only the y-component adds up. | |||
=== Step 3: Integrate === | |||
We now integrate from <math>-L/2</math> to <math>+L/2</math>: | |||
<math> | |||
E_y = \frac{\lambda y}{4\pi\varepsilon_0} \int_{-L/2}^{L/2} \frac{dx}{(x^2 + y^2)^{3/2}} | |||
</math> | |||
This integral has a standard solution: | |||
<math> | |||
E_y = \frac{\lambda}{4\pi\varepsilon_0 y} \cdot \left( \frac{L/2}{\sqrt{(L/2)^2 + y^2}} \right) | |||
</math> | |||
Final result: | |||
<math> | |||
E_y = \frac{Q}{4\pi\varepsilon_0 L y} \cdot \left( \frac{L/2}{\sqrt{(L/2)^2 + y^2}} \right) | |||
</math> | |||
[[Image:ElectricFieldVectorFromRod.png|600px|center|thumb|Vector from a rod element to the observation point]] | |||
=== Step 4: Check the Result === | |||
* '''Units''': The result has units of N/C, as expected. | |||
* '''Direction''': The field points away from the rod if <math>Q > 0</math>, and toward the rod if <math>Q < 0</math>. | |||
* '''Limiting Behavior''': As <math>y \gg L</math>, the result simplifies to the electric field of a point charge: | |||
<math> | |||
E \approx \frac{Q}{4 \pi \varepsilon_0 y^2} | |||
</math> | |||
=== Computational Models === | === Computational Models === |
Revision as of 16:44, 13 April 2025
Milo Karnes, Spring 2025
The Main Idea
In earlier studies, we learned about the electric field created by a point charge. However, in the real world, charges are often spread out over objects with shape and structure. One common example is a uniformly charged rod. To determine the electric field from such an object, we divide the rod into many infinitesimally small charge segments, treat each as a point charge, and integrate their contributions.
The key idea is to approximate the rod as a continuous line of charge using the principle of superposition. We consider the symmetry of the setup to simplify the problem and focus on the components of the electric field that don't cancel out.
The process of finding the electric field from a charged rod involves four main steps:
- Model the rod as many small charge elements and draw the electric field vector [math]\displaystyle{ \Delta \vec{E} }[/math] from a single element.
- Use symmetry to argue which components cancel and which remain.
- Integrate the contributions from all elements to find the net electric field.
- Verify that the result makes physical sense (units, direction, and limiting behavior).
A Mathematical Model
To calculate the electric field of a uniformly charged rod, we treat the rod as a continuous distribution of charge. Let the rod have total length [math]\displaystyle{ L }[/math] and total charge [math]\displaystyle{ Q }[/math], centered along the x-axis. The observation point is located on the y-axis a distance [math]\displaystyle{ y }[/math] above the center.
Step 1: Break the Rod into Pieces
We divide the rod into tiny segments of length [math]\displaystyle{ dx }[/math]. Each segment behaves like a point charge: [math]\displaystyle{ dq = \lambda \, dx }[/math], where [math]\displaystyle{ \lambda = \frac{Q}{L} }[/math] is the linear charge density.
Each [math]\displaystyle{ dq }[/math] contributes a small electric field [math]\displaystyle{ d\vec{E} }[/math] at the observation point.
Step 2: Write the Field Expression for One Piece
Using Coulomb’s Law, the electric field contribution from one element is:
[math]\displaystyle{ d\vec{E} = \frac{1}{4\pi\varepsilon_0} \cdot \frac{dq}{r^2} \cdot \hat{r} }[/math]
The vector from the source to the observation point is:
[math]\displaystyle{ \vec{r} = \langle 0, y \rangle - \langle x, 0 \rangle = \langle -x, y \rangle }[/math]
So:
[math]\displaystyle{ r = \sqrt{x^2 + y^2} \quad \text{and} \quad \hat{r} = \frac{\langle -x, y \rangle}{\sqrt{x^2 + y^2}} }[/math]
Putting it all together:
[math]\displaystyle{ d\vec{E} = \frac{1}{4\pi\varepsilon_0} \cdot \frac{\lambda \, dx}{(x^2 + y^2)} \cdot \frac{\langle -x, y \rangle}{\sqrt{x^2 + y^2}} = \frac{\lambda}{4\pi\varepsilon_0} \cdot \frac{\langle -x, y \rangle \, dx}{(x^2 + y^2)^{3/2}} }[/math]
Note: The x-components of the field cancel due to symmetry. Only the y-component adds up.
Step 3: Integrate
We now integrate from [math]\displaystyle{ -L/2 }[/math] to [math]\displaystyle{ +L/2 }[/math]:
[math]\displaystyle{ E_y = \frac{\lambda y}{4\pi\varepsilon_0} \int_{-L/2}^{L/2} \frac{dx}{(x^2 + y^2)^{3/2}} }[/math]
This integral has a standard solution:
[math]\displaystyle{ E_y = \frac{\lambda}{4\pi\varepsilon_0 y} \cdot \left( \frac{L/2}{\sqrt{(L/2)^2 + y^2}} \right) }[/math]
Final result:
[math]\displaystyle{ E_y = \frac{Q}{4\pi\varepsilon_0 L y} \cdot \left( \frac{L/2}{\sqrt{(L/2)^2 + y^2}} \right) }[/math]
Step 4: Check the Result
- Units: The result has units of N/C, as expected.
- Direction: The field points away from the rod if [math]\displaystyle{ Q \gt 0 }[/math], and toward the rod if [math]\displaystyle{ Q \lt 0 }[/math].
- Limiting Behavior: As [math]\displaystyle{ y \gg L }[/math], the result simplifies to the electric field of a point charge:
[math]\displaystyle{ E \approx \frac{Q}{4 \pi \varepsilon_0 y^2} }[/math]
Computational Models
Finding the Electric Field from a Rod with Code While computation done by hand does have its merits, and certainly was the methodology used when these ideas were conceived, there are much more efficient, powerful, and most importantly, pretty ways to go about finding and showing the electric field. This is of course referring to computers, specifically computers running glowscript code in the case of Physics 2212. With that idea in mind, here are some demonstrations of said methodology:
This is some code that you can run which shows the electric field vector
at a given distance from the rod along its length. The rod is shown as a
series of green balls to help emphasize that when using the numerical
integrations mentioned on this page, you are measuring the field produced
by discrete parts of the rod being analyzed. At each point of analysis,
eight field arrows are shown so as to visualize the electric field.
Notice the edge-effects of the electric field of the rod. For reasons discussed above, if we used the long rod approximation (L>>d), these effects would be negligible.
For a more precise model, the two links below lead to code that generates a forty element line of charge with given magnitude, and length, and then iterates vectors representing the electric field in the space all around said line of charge. Note that the vectors are small, but for the positively charged rod, they lead radially outward, and for the negative, radially inward. This is due to the fact that the rod(s) are treated as lines of positive or negative charge, and the electric field behaves as such.
Try zooming in and out! You can really see the symmetry of the field far out, and the edge effects when zoomed in.
Examples
Although this is not a very difficult topic, some reasonably difficult conceptual questions can be asked about it.
Simple

Middling

Difficult

Connectedness
The electrical field of a charged rod has many real world applications. Even within other areas of physics, you can extend what we know about a charged rod to find out the electric field of other objects. For instance, a ring is also just a rod that is bent in a circle. Further, a disk is a collection of many concentric rings. This method of thinking can lead us to some very interesting derivations for physics.
Even if we are to only think about real life, rods exist everywhere. Since everything has an electrical field, any rod or even simply cylindrical shape or anything made of rods is an application. A very useful example that also comes up in physics is the concept of wires. Wires are, in essence, thin rods in which electrons can flow through. This categorizes them as a charged rod. Wires are not only an important subject in physics, but also real life as the device you are reading this on probably needed a wire for charge! What happens inside of the wire is important, but we also what happens outside of it is equally important. If we consider static electricity, a misunderstanding or a failure to account for static could be detrimental at a large scale as static can very easily cause malfunctioning to anything that uses wires to operate.
I, the most recent editor of this page, see the use of the electric field of charged rods as a CS major constantly for a similar reason: wires. Computer Science and any form of software really only exists with accompanying hardware. Almost always, the connection between software and hardware is wires and the electrical pulses that serve as the data that travels along the wire. Especially as someone looking into robotics, it is important to understand the outward effect that wires have because static can be very tricky if you don't know how to handle it!
History
Physicists and scientists make use of electric fields and charged objects all the time. Many times, we may need to know which objects are contributing how much charge in certain areas. Charged objects may attract or repel (depending on the signs of their charge), so we often need to know how objects will interact with each other based on their charges. The phenomenon of this interaction, or electric force between charged particles, was finally confirmed and stated as a law in 1785 by French physicist Charles-Augustin de Coulomb, hence "Coulomb's Law."
See also
The equation for the electric field of a charged rod was derived from the equation for the electric field of a charged particle. See the article "Electric Field" for more information.
Further Reading
The page on electric fields: Electric Field
External Links
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elelin.html
http://online.cctt.org/physicslab/content/phyapc/lessonnotes/Efields/EchargedRods.asp
http://dev.physicslab.org/Document.aspx?doctype=3&filename=Electrostatics_ContinuousChargedRod.xml
References
https://www.youtube.com/watch?v=BBWd0zUe0mI
(For the above reference, the textbook's method is followed in that the charge distribution was left undefined, and assumed to be constant)
Chabay and Sherwood: Matter and Interactions, Fourth Edition, Chapter 15