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Rosengrip

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## Homework Statement

We have a charged ball with charge

**e**and mass

**m**hanging from the side of the charged cylinder with radius

**R**and surface charge density of [itex]\sigma[/itex].

The string to which the ball is attached is of length

**L**

Find a relation between charge

**e**on the ball and the angle [itex]\varphi[/itex] between cylinder and string.

http://www.shrani.si/?2J/AQ/40Q2N2jg/cyllinder.png

## Homework Equations

Gauss' law for calculating electric field where ball is.

## The Attempt at a Solution

Gauss law for electric field around cylinder gives us:[itex]E=\frac{\lambda}{2\pi \cdot x\cdot \epsilon_0}[/itex]

Considering linear density [itex]\lambda[/itex] equals [itex]\sigma[/itex]2PI*R (R is radius of cylinder), we get the expression:

[itex]E=\frac{\sigma R}{x \epsilon_0}[/itex]

[itex]\sigma[/itex] is surface charge density.

where x equals distance from centre of the cylinder to ball position, meaning [itex]x = R + Sin[\phi]\cdot L[/itex]

After drawing a free body diagram and eliminating the force of string, we have the electric force and force of gravity left.

Electric force thus equals: [itex]F_E=\frac{e\sigma R}{\epsilon_0 (R+Sin[\phi] L)}[/itex]

We get the followting expression:[itex]Tan[\phi] = \frac{e\sigma R}{\epsilon_0 m g (R+Sin[\phi] L)}[/itex]

Trying to solve this for phi is pretty painful, even Mathematica can't really properly do it (even for substituting Tan with Sin for small angles)

Is there any other way for showing a relation between charge and angle?

Any help would be greatly appreciated.

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