Why is this equation from Purcell's EM textbook correct?

In summary, the conversation discusses an equation on page 31 of Purcell's EM textbook that calculates the force on a unit area. The formula is originally written as F = Ɛ0∫E1E2 E dE = 2/ Ɛ0 (E2^2 - E1^2), but is then simplified to F = 1/ Ɛ0 (E1 + E2)σ. There is a discrepancy with the coefficient, which the author believes should be Ɛ0/2 instead. The conversation also mentions that the book may have printed an older version of the formula, as the newer version in the 3rd edition shows the correct coefficient.
  • #1
Leo Liu
353
156
Homework Statement
.
Relevant Equations
.
I saw the following equation on page 31 in Purcell's EM textbook.
$$F=\epsilon_0\int_{E_1}^{E_2} E\, dE=\frac 2 {\epsilon_0} (E_2^2-E_1^2)$$
Here, F is the force on a unit area.
And then he claims that since ##E_2-E_1=\sigma/\epsilon_0##, the equation can be further simplified to
$$F=\frac 1 {\epsilon_0}(E_1+E_2)\sigma$$

However, I think the correct coefficient in the last part of first equation should be the inverse of what it is now (##\frac{\epsilon_0}{2}## instead), as only then can I obtain the second expression. I have no idea why the author wrote the equation this way. To give it a little bit of background, dE is the change in the electric field of a thin layer of a charged sphere, E1 is the the electric field inside, and E2 is the electric field outside. Could someone explain?
 
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  • #2
yes, at least viewing from a purely mathematical view (integrating ##\int x dx=\frac{x^2}{2}## that should have been ##\frac{\epsilon_0}{2}##. Also from a dimensional analysis point of view ##\epsilon_0## just cannot be on the denominator, the units won't match then.
 
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  • #3
Leo Liu said:
Homework Statement:: .
Relevant Equations:: .

I saw the following equation on page 31 in Purcell's EM textbook.
$$F=\epsilon_0\int_{E_1}^{E_2} E\, dE=\frac 2 {\epsilon_0} (E_2^2-E_1^2)$$
The formula (1.48 in the book) is in fact
##\frac F A = … = \frac {\epsilon_0}{2}(E_2^2 – E_1^2)##
Check your book!
 
  • #4
Steve4Physics said:
The formula (1.48 in the book) is in fact
##\frac F A = … = \frac {\epsilon_0}{2}(E_2^2 – E_1^2)##
Check your book!
1628347234618.png

Here is a picture of my book. This book is probably an older edition which is authorized to be printed in China by the original publisher.
 
  • #5
Leo Liu said:
;View attachment 287228
Here is a picture of my book. This book is probably an older edition which is authorized to be printed in China by the original publisher.
I guess this is an error in earlier editions which has now been corrected. The 3rd edition (bottom of page 31) shows this:
force per unit area.jpg
 
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1. What is the significance of Purcell's EM textbook equation?

Purcell's EM textbook is a widely used and respected resource for studying electromagnetic theory. The equations presented in the textbook are derived from fundamental principles and have been rigorously tested and verified, making them a reliable and accurate representation of electromagnetic phenomena.

2. How can I be sure that Purcell's EM textbook equation is correct?

The equations in Purcell's EM textbook have been extensively reviewed and tested by experts in the field of electromagnetic theory. Additionally, these equations have been used in countless experiments and applications, further validating their accuracy and correctness.

3. Are there any limitations to Purcell's EM textbook equation?

Like any scientific theory or equation, Purcell's EM textbook equation has its limitations. It may not accurately describe certain complex electromagnetic phenomena or may require additional assumptions or modifications in certain scenarios. It is important to understand the context and assumptions behind the equation before applying it.

4. How can I apply Purcell's EM textbook equation in my research or experiments?

Purcell's EM textbook equation can be applied in various research and experimental settings to study and understand electromagnetic phenomena. However, it is crucial to have a thorough understanding of the equation and its limitations before using it in any application.

5. Can I use Purcell's EM textbook equation in real-world applications?

Yes, Purcell's EM textbook equation has been successfully used in numerous real-world applications, such as designing electronic devices, analyzing electromagnetic fields, and predicting the behavior of electromagnetic waves. However, it is important to carefully consider the assumptions and limitations of the equation in each specific application.

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