Show simplification of dipole's electric field for certain case

In summary, the conversation discusses how to show that the equation for the electric field of a dipole breaks down to E = Q/(2π ##ϵ_0## ##r^2##) * ##i^##. The approach involves examining the x-components of the electric field and using the Pythagorean theorem to define r. However, there may be some typos in the work presented, resulting in discrepancies in the final answer. The other person in the conversation asks for clarification on this issue.
  • #1
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Homework Statement


I'm just trying to show that the equation in the attached image becomes E = Q/(2π ##ϵ_0## ##r^2##) * ##i^##, where ##i^## is i-hat, the unit vector for when r is the smallest that it could be, which is when r = d/2.

I'm doing this with reference to problem 1.2 (and I'm including problem 1.1 so that you have the background information for problem 1.2).

Homework Equations


The equation in the image as well as E = Q/(2π ##ϵ_0## ##r^2##) * ##i^##.

The Attempt at a Solution


My attempt is attached as MyAttempt.pdf.

Is it a coincidence that I'm getting double the value shown in problem 1.2?

Any help in getting to show that the vector equation for the electric field of a dipole breaks down to E = Q/(2π ##ϵ_0## ##r^2##) * ##i^## would be greatly appreciated!
 

Attachments

  • ElectricField_ForElectricDipole_Equation.jpg
    ElectricField_ForElectricDipole_Equation.jpg
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  • MyAttempt.pdf
    41.4 KB · Views: 150
  • 1.1.jpg
    1.1.jpg
    55.8 KB · Views: 356
  • 1.2.jpg
    1.2.jpg
    49.7 KB · Views: 360
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  • #2
You are approaching this well, but is easier to why this result comes if you examine it like this.

The x-components of the electric field add, so we say that,

[tex]E_t = E_1 + E_2\\
E_1 = \frac{1}{4\pi\epsilon_o}\frac{Q}{r^2}sin\theta[/tex]

Which is the same as E_2 because both value of sines are positive. Even though we have a specific value for what r should be, to justify our solution we will define r using the Pythagorean theorem. So the total electric field then becomes.

[tex]E_t = \frac{2}{4\pi\epsilon_o}\frac{Q}{y^2 + \frac{d}{2}}sin\theta\\
sin\theta = \frac{d}{2}{r}\\
E_t =\frac{1}{4\pi\epsilon_o}\frac{Qd}{r^3}\\
E_t = \frac{Qd}{4\pi\epsilon_o}\frac{8}{d^3}\\
E_t = \frac{Q}{2\pi\epsilon_od^2}[/tex]

Then remember that your d value is actually
[tex] d^2 = \frac{d^2}{4} = r^2[/tex]

This should give you the answer that you are searching for.
 
  • #3
Thanks for your answer.

Isn't what I got in the PDF of my work (in my opening/first post of this thread) ##E_t##?

It seems that you made a few typos, so in an attempt to overlook the typos, I think your work results in ##E_t## = Q/(2π ##ϵ_0## ##r^2##), such that ##E_1## = ##E_2## = Q/(4π ##ϵ_0## ##r^2##).

I'm mentioning this because my ##E_t## seems to be double yours (which in turn is double ##E_1## and ##E_2##, respectively – such that my ##E_t## is four times ##E_1## and ##E_2##, respectively), so while I think I see what you're trying to say, I'm still not getting the algebra to confirm it.

Could you please elaborate on this situation?
 

1. What is a dipole's electric field?

A dipole's electric field is a type of electric field that is created when two equal and opposite charges are separated by a small distance. This results in a field that is strongest at the poles and weakest at the center of the dipole.

2. How is a dipole's electric field simplified?

A dipole's electric field can be simplified by using the dipole moment, which is a vector quantity that represents the strength and direction of the dipole. This allows for a more concise representation of the electric field compared to the individual contributions of each charge.

3. When is it necessary to simplify a dipole's electric field?

It is necessary to simplify a dipole's electric field when analyzing the behavior of the field in a specific case or scenario. Simplification can make calculations and interpretations of the field easier and more accurate.

4. Can the electric field of a dipole be simplified for all cases?

No, the simplification of a dipole's electric field is specific to certain cases. In some cases, the individual contributions of each charge may need to be considered in order to accurately describe the field.

5. What are some examples of cases where simplification of a dipole's electric field is useful?

Simplification of a dipole's electric field is useful in cases such as when analyzing the behavior of ions in a solution, or when studying the electric field of molecules in a polar solvent. It can also be helpful in understanding the behavior of electric dipoles in materials used for electronic devices.

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