\$\delta n \cdot x + n \cdot \delta x~\$ equation of shrunk coil with incremented number of turns per unit length

Question

The very long coil exists with the following information .

$$ a:=\text{radius of the circle of the coil} $$

$$ n:=\text{number of turns of the coil per unit length} $$

$$ x:=\text{lenght of the coil(not a length of the wire )} $$

$$ N:=nx ~~ \leftarrow~~ \text{total number of turns of the coil} $$

$$ I:=\text{current which is to be flown to the coil} $$

We want to know the force of shrink of the coil .

$$ H=nI ~~ \leftarrow~~ \text{magnetic field inside the coil } $$

$$ \Phi=\underbrace{\mu_{0}H}_{\text{Wb}/\text{m} ^{2} } \cdot \underbrace{\pi a ^{2}}_{\text{m}^{2} } \cdot \underbrace{nx}_\text{turns} ~~ \leftarrow~~ \text{interlinkage mangtic flux} $$

$$ U_{\text{m} } = \frac{1}{2} I \Phi_{} ~~ \leftarrow~~ \text{energy} $$

\$~ \delta x ~~ \leftarrow~~ \text{extent of length of shrink} ~\$

\$~ \delta n ~~ \leftarrow~~ \text{incremented number of turns per unit length as shrink is done } ~\$

What I can't get currently is the below equation .

$$ \delta n \cdot x + n \cdot \delta x =0 $$

What should I consider first?

Answer 1

I don't know what class you are taking or if this is self-education (which I applaud, if so.) And I certainly have zero knowledge of any context you've acquired before reaching this point. But I can read equations.

I think the idea here is that \$\partial\:\Phi=0\$ (the flux itself isn't changing.) Since the flux isn't changing, it follows that neither is \$\Phi_{_0}=\mu_{_0}H\pi a^2\$ (which, of course, is the static flux under consideration.)

So, given \$\Phi=\Phi_{_0}\cdot nx\$, and applying the derivative operator:

$$\begin{align*}D\:\Phi &= D\:\left\{\Phi_{_0}\cdot nx\right\} \\\\ &= \Phi_{_0}\cdot D\:\left\{ nx\right\} \\\\ &=\Phi_{_0}\cdot \left[D\:\left\{ n\right\}\cdot x+D\:\left\{ x\right\}\cdot n\right] \\\\ \partial\:\Phi&= \Phi_{_0}\cdot \left[\partial n\cdot x+\partial x\cdot n\right] \end{align*}$$

Given the static situation, you know that \$\partial\:\Phi=0\$, so:

$$\begin{align*} 0&= \Phi_{_0}\cdot \left[\partial n\cdot x+\partial x\cdot n\right] \end{align*}$$

And since \$\Phi_{_0}\$ cannot be assumed to be zero, it follows that the only remaining factor must be zero:

$$\begin{align*} 0&= \partial n\cdot x+\partial x\cdot n \end{align*}$$