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Category:Computer-related introductions in 2010Q:

Source for finding energy loss in an isolator

The following example is from page 3 of this book

The energy loss in the isolator can be found as the work done on a magnet at the end of the path, so by Hooke’s law we have $F = -k x$. We find the change in energy as the magnet is moved along the path by applying Stoke’s law to the magnet at the end of the path so that the integral is done by parts:
$\Delta E = \int_{0}^\infty F (x) \; dx$
$= \left (\frac{F}{ -k}\right )_{0}^{\infty} + k x_1 – \int_{0}^{x_1} F(x) dx$
$= \left (\frac{F}{ -k}\right )_{0}^{\infty} + k x_1 – \int_{0}^{x_1} -kx \; dx = kx_1$

My questions are as follows:
Is this a general method to find the energy loss when an object is moving through a path when the net force is constant $F$?
If this method is valid, where can I find a more rigorous proof of it?

A:

The main idea is correct, but the proof is not. You should find your energy balance in the form
$$\Delta E=\int_{x_0}^{x_1} \vec{F}\cdot d\vec{s},$$
where $\vec{F}$ is the effective force on the object.
The point is that the force on the object at $x$ is $\vec{F}=-k\hat{x}$ by the parallel conductor idea, so if you move the object from $x_0$ to $x_1$ then the work \$\int_{x_0}^{x_1} \vec{F}\cdot d\vec{s}
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