How to Derive the Expression for \(\delta W = -E dP\)?

  • Thread starter Thread starter appmathstudent
  • Start date Start date
appmathstudent
Messages
6
Reaction score
2
Homework Statement
This is exercise 3-12 form Sears and Salinger Thermodynamics : Show that $$\delta W = -E dP$$ by calculating the work necessary to charge a parallel plate capacitor containing a dielectric.
Relevant Equations
$$W = U = \frac{q^2}{2C}$$
$$ C = \frac{\kappa \epsilon_0 A}{d}$$
$$P = qd $$ (dipole moment of slab)
$$ E = \frac{q}{\epsilon_0 \kappa A}$$
$$W = U = \frac{q^2}{2C} =\frac{q q d}{2 \kappa \epsilon_0 A} = \frac{E P}{2}$$

Then , since E is constant we have that :

$$\delta W = \frac{dW}{dP} dP = \frac{E}{2} dP$$.

My question is how can I make this 2 on the denominator disappear in order to obtain the required expression ?

ps : In the book (Chapter 3 page 67) he mentions that $$\delta W$$ is the work when $$E$$ is changed in a dielectric slab.
 
Last edited:
Physics news on Phys.org
dP=d(qD)=(dq) D+ q (dD)
where D is distance between the capacitor plates introduced to distinguish it with differential d.
Which is your case changing charge or changing distance or the both ?
 
Last edited:
I think q is changing, since the work is done to change E.
 

Attachments

  • 20211002_125815.jpg
    20211002_125815.jpg
    30 KB · Views: 164
appmathstudent said:
I think q is changing, since the work is done to change E.
So you are saying E is not constant during the charging process.
 
appmathstudent said:
I think q is changing, since the work is done to change E.
q is changing, d is constant so
dP=d(dq)
\frac{dW}{dP}=\frac{1}{d}\frac{dW}{dq}
Try it.
 
  • Like
Likes appmathstudent
Hi, I had an exam and I completely messed up a problem. Especially one part which was necessary for the rest of the problem. Basically, I have a wormhole metric: $$(ds)^2 = -(dt)^2 + (dr)^2 + (r^2 + b^2)( (d\theta)^2 + sin^2 \theta (d\phi)^2 )$$ Where ##b=1## with an orbit only in the equatorial plane. We also know from the question that the orbit must satisfy this relationship: $$\varepsilon = \frac{1}{2} (\frac{dr}{d\tau})^2 + V_{eff}(r)$$ Ultimately, I was tasked to find the initial...
The value of H equals ## 10^{3}## in natural units, According to : https://en.wikipedia.org/wiki/Natural_units, ## t \sim 10^{-21} sec = 10^{21} Hz ##, and since ## \text{GeV} \sim 10^{24} \text{Hz } ##, ## GeV \sim 10^{24} \times 10^{-21} = 10^3 ## in natural units. So is this conversion correct? Also in the above formula, can I convert H to that natural units , since it’s a constant, while keeping k in Hz ?
Back
Top