Question

I need help solving for the charge “q”. My professor says it’s just algebra but it seems to be challenging. I added photos of the lab and a free body diagram for reference.

Estimating the Charge on a Simple Electroscope Questions one poses to nature in physics are answered through measurement and modeling. The answer has uncertainties not only due to measurements, but how good the model is. This lab is constructed not as a cookbook exercise, but to mimic how research really is done on a very simple scale. You are presented with on a simple "scotch tape" electroscope, where ripping the tape from a surface results in charge being deposited on the tapes. Then you will calculate the charge using your knowledge of models we discuss in class, a simple model and a refined model. There will be uncertainties in each model due to the measurements themselves, but you will also obtain the difference in the calculation using the simple and refined model. An electroscope used in demonstrations often looks like the figure below 1,小4, charged Fig.1- a demo electroscope Two metal lcaves (gold usually) are attached to a metal post, which in turn is attached to a metal ball. Charge is transferred from some insulator that has been charged up (like a plastic stick rubbed with fur), and the leaves separate as the charge redistributes itself all along the metal surface A simple electroscope can be also be made from two pieces of scotch tape. Tape will get charge from a surface it is stuck to if it is ripped off that surface quickly. Put together the two pieces of tape and you have something looking like Fig.2. The charge of course doesn't spread out along the tape- it is an insulator, but it should be distributed reasonably uniformly over the tape surface. The goal of the lab is to calculate and estimate of the following: the total charge, charge density and number of electrons per molecule assuming it is made of cellulose acetate (CHidos).

Answer 1

Let us consider model 1

here q1 =q2 =q and l1=l2=L and m1 =m2=m

forces on any point charge a on mass m are:

mg vertically downwards

coloumb charge towards on other charge in horizontal direction

and tension T in the string

let $\theta$ be the angle each makes with vertical

kq^2/r^2 - T sin$\theta$ =0 Where k=$K= 1/4pi\varepsilon_{0})$‘’’’

Mg - T Cos$\theta$ = 0

tan$\theta$ = kq^2/(r^2Mg)

let x be distance between 2 charges

tan$\theta$ =( x/2)/ L

$Kq^2/(2r^2Mg)= (x/2L)$

$q = sqrt (2r^2Mgx/2Lk)$

In the second model if we consider the entire strip is considered to have uniform surface change density $\sigma$

Field at any point $E=\sigma /2\epsilon _{0}$

force on Q= $Q\sigma /2\epsilon _{0}$

$Q\sigma /2\epsilon _{0}$ = T sin (theta)

Mg = T cos (thetA)

tan(theta) = Q (sigma) / (2epsilon Mg) = (x/2L)

Q = 2x( epsilon) Mg/ 2L (sigma)

so in model 2 I assumed the charge is spread over entire area of the strip and that strip is quite big compared to Charge Q