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Question

A thin disk with a circular hole at its center, called an anmulus, has inner radius $R_1$ and outer radius $R_2$. The disk has a uniform positive surface charge density $\sigma$ on its surface. (c) Show that at points on the $x$-axis that are sufficiently close to the origin, the magnitude of the electric field is approximately proportional to the distance between the center of the annulus and the point. How close is "sufficiently close"?

Solution

VerifiedAnswered 11 months ago

Answered 11 months ago

Step 1

1 of 2**c)** If $x<<R_{_1}$ and $x<<R_{_2}$ then the expression in part (b) is reduced to
$$E=\dfrac{\sigma x}{2\epsilon_{_o}}\left[\dfrac{1}{R_{_1}}-\dfrac{1}{R_{_2}} \right]$$
The only variable on previous expression is $x$ and the electric field is proportional to it at very close distance $x$ .
The distance should be closed at the point we can ignore it with the inner and outer radius or the inner and outer radius should be so large .

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