The force between two conductors (Duffin)

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SUMMARY

The discussion centers on the derivation of the force between two conductors as presented in "Electricity and Magnetism" by Duffin, specifically in section 5.7. The participant grapples with the equation for electrical energy stored, particularly the transition from dU = 0.5QdV to dU = -0.5V^2 * dC. The confusion arises from the treatment of voltage as constant versus changing, leading to a proposed correction of dU = 0.5 d[V^2 * C]. Ultimately, the participant confirms their understanding by deriving the correct expression, which reconciles the negative sign in the force equation.

PREREQUISITES
  • Understanding of electrical energy concepts, specifically in the context of capacitors.
  • Familiarity with the principles of electromagnetism as outlined in Duffin's textbook.
  • Knowledge of calculus, particularly differentiation and the application of derivatives in physics.
  • Basic understanding of the relationship between charge, voltage, and capacitance.
NEXT STEPS
  • Study the derivation of the force between conductors in more detail using "Electricity and Magnetism" by Duffin.
  • Learn about the implications of changing voltage and capacitance in capacitor systems.
  • Explore the concept of energy stored in capacitors and its derivation from fundamental principles.
  • Investigate the role of negative signs in physics equations and their significance in energy calculations.
USEFUL FOR

Students of electromagnetism, particularly first-year physics students, educators teaching electricity and magnetism, and anyone seeking to deepen their understanding of capacitor behavior and energy relationships in electrical systems.

Smith987
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Homework Statement


I have been reading electricity and magnetism by Duffin in order to understand my first year EM course. So far it has been straightforward- however, I've hit a wall when it derives the force between 2 conductors (p.117 section 5.7 if you have the book!). The book says:


Homework Equations


Force along x = - dU/dx with Q held constant, where U=electrical energy stored(I accept this)

Elementary increment dU= 0.5QdV (i accept)
=0.5 Q^2 * d(1/c) (I accept)
=-0.5 V^2 * dC.

This I do not accept, as in the first equality, it implies V is changing (dV) yet in the last equality V is constant? I'm also unsure as to where the minus sign cam from but I assume it was a misprint.


The Attempt at a Solution



The only answer I can come up with is that the equation was meant to be
dU= 0.5 d[ V^2 * C]. But if I use this for the example of a parallel plate capacitor I get a sign error.

Please help!
 
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I think I solved the problem- when you consider dU as 0.5 *d(V^2 * C), this is equal to

0.5 * (V + dV)^2 * (C + dC) - 0.5*V^2 * C, which by using dv/dc= -Q/(C^2) leaads to the answer given. And pleasingly it gave me that annoying negative sign :)

I would still like a pat on the back though.
 

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