# Basic electrostatics problem; analytical solution?

• Taulant Sholla
In summary: I get that? which, if correct, then comes to ½ sin(2θ) = kq2/4L2mgwhich, if correct, then comes to 1 - sin(2θ) = -kq2/4L2mg
Taulant Sholla

## Homework Statement

For the system given, both objects have the same charge and same mass (both given). I'm also given string length, L. I need to solve for θ.

## Homework Equations

Coulomb's Law, W=mg

## The Attempt at a Solution

Using simple equilibrium force analysis (with weight, tension, and electrostatics forces), I get:

Is there a way to solve for θ analytically, or do I have to find a graphical solution?
Thank you!

It might be manipulated into the form of a cubic equation. Cubics have a closed form solution for their roots. It'll involve a couple of changes of variables I think. Try expressing cos(θ) in terms of sin(θ), and then call x = sin(θ) so you're working with x rather than trig.

Taulant Sholla said:

## Homework Statement

View attachment 112497
For the system given, both objects have the same charge and same mass (both given). I'm also given string length, L. I need to solve for θ.

## Homework Equations

Coulomb's Law, W=mg

## The Attempt at a Solution

Using simple equilibrium force analysis (with weight, tension, and electrostatics forces), I get:
View attachment 112499
Is there a way to solve for θ analytically, or do I have to find a graphical solution?
Thank you!

The equation ##\sin^3(\theta)/\cos(\theta) = a## is not difficult to solve analytically. For ##a > 0## we have ##0 < \theta < \pi/2##, so ##\cos(\theta) = \sqrt{1 - \sin^2(\theta)} > 0##. Therefore, the new variable ##x = \sin^2(\theta)## obeys the cubic equation ##x^3/(1-x) = a^2##. The exact solution of this cubic is not too complicated or difficult to work with. From ##x## we can recover ##\theta = \arcsin(\sqrt{x})##.

Last edited:
I get that
sin(θ) cos(θ) = kq2/4L2mg
?
which, if correct, then comes to

½ sin(2θ) = kq2/4L2mg

I believe this is incorrect. I'm pretty sure of the solution I posted, since it does agree with computational results.

Thank you so much!
I went down this path and saw the enormously complexity of solving cubics, which is - I guess - why we're taught to use graphical solutions methods instead.

Ray Vickson said:
The equation ##\sin^3(\theta)/\cos(\theta) = a## is not difficult to solve analytically. For ##a > 0## we have ##0 < \theta < \pi/2##, so ##\cos(\theta) = \sqrt{1 - \sin^2(\theta)} > 0##. Therefore, the new variable ##x = \sin^2(\theta)## obeys the cubic equation ##x^3/(1-x) = a^2##. The exact solution of this cubic is not too complicated or difficult to work with. From ##x## we can recover ##\theta = \arcsin(\sqrt{x})##.

## 1. What is electrostatics and why is it important?

Electrostatics is the study of electric charges at rest. It is important in understanding and explaining various phenomena, such as static electricity, electric fields, and electric potential. It also plays a crucial role in the design and function of electronic devices.

## 2. What are some common types of electrostatics problems?

Some common types of electrostatics problems include finding the electric field or potential at a given point due to a known charge distribution, calculating the force between two charged objects, and determining the capacitance of a system.

## 3. What is the analytical solution method for solving electrostatics problems?

The analytical solution method involves using mathematical equations and formulas to solve for the desired quantity in an electrostatics problem. This often involves applying the principles of Coulomb's law, Gauss's law, and the superposition principle.

## 4. How can I check if my analytical solution for an electrostatics problem is correct?

One way to check the correctness of an analytical solution is to use the method of dimensions, which involves checking if the units of the answer obtained match the expected units. Another way is to compare the solution to the results obtained through numerical methods or experimental data.

## 5. Are there any common mistakes to avoid when solving electrostatics problems?

Some common mistakes to avoid include using incorrect sign conventions for charges, forgetting to take into account the direction of the electric field, and not considering the influence of nearby charges or conductors. It is also important to double-check all calculations and units to ensure accuracy.

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