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

The very long coil exists with the following information .

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

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

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

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

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

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

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

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

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

$$\~ \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?

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*}