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- Homework Statement
- An arc with radius 𝑅 and subtending an angle 𝜃 is suspended by two wires as in

Figure. It has a charge density per unit length 𝜆, and a total mass 𝑚. The wires

have a maximum tension before they break of 𝑇max. We add a charge with charge

𝑞 fixed at the geometric centre of the arc. What is the maximum value that we can

give to the charge 𝑞 before the wires snap and the arc falls? Remember to include

gravity.

[Hint: you need to calculate the balance of the forces. Tensions of the wires and

gravity are going to be easy, but the electric force due to the repulsion between the

charge and the arc are a bit more difficult (you need to do an integral).]

- Relevant Equations
- E = kq/R^2, Weight = mg

I need to account for tension, weight, and repulsion.

For the tension, I can draw the x and y component of Tmax and see that the x components of the 2 tensions Tmax will cancel out, and there are 2 y components of the Tmax to factor in.

Weight is just F = mg, where g is acceleration due to gravity.

The direction of Fgravity = downwards while Ftension = upwards. The moment Fupwards =/= Fdownwards, the strings will snap. Frepulsion is downwards, so Forces acting downwards are gravity + repulsion, while Forces acting upwards is tension.

$$ dE = k* \frac{dq}{R^2} $$

Solving for dE, I get $$ dE = k* \frac{𝜆 d𝜃}{R} $$

since we know S = R𝜃

However, I have trouble solving for E, where

$$ E = 2 * \int cos(𝜃/2) \, dE $$ from 0 to infinity.

I am unable to solve this part because cos(𝜃/2) does not converge when 𝜃 goes to infinity. May I know where I went wrong?