Metal sphere on a thread in a horizontal electric field

In summary, a metal sphere hanging on a thread in a horizontal electric field at a 45 degree angle can be expressed as ##\mg=q \cdot E \cdot \tan(45)##, and after removing 40% of the charge, the angle becomes ##\Theta_2=59^{\circ} 04'##.
  • #1

Homework Statement



Charged metal sphere hanging on an isolated thread of negligible mass is put in a homogeneous horizontal electric field so that the thread makes a 45 degree angle with the el. field. What angle does the thread with the sphere close with the el. field after we remove 40% of the charge from the sphere?

Homework Equations



1. ##F_g=mg##
2. ##F_g=T \cdot \sin(45)##
3. ##F_{el}=T \cdot \cos (45)##

I have provided image of force diagram.

The Attempt at a Solution



I can substitute ##F_g## from equation 1 into equation 2 and get:
##mg=T \cdot \sin(45)##

Then I find expression for T in equation 3:
##T=\frac{F_{el}}{\cos (45)}##

and substitute it into second equation:
##mg=\frac{F_{el}}{\cos (45)} \cdot \sin(45)##
##mg=F_{el} \cdot \tan(45)##

I can express electric force as a product of electric field and charge of sphere:
##mg=q \cdot E \cdot \tan(45)##

I haven't gotten further than this. How do I show what happens with an angle when the charge is reduced 40%?
 

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  • #2
steroidjunkie said:
I can express electric force as a product of electric field and charge of sphere:
##mg=q \cdot E \cdot \tan(45)##
This equation looks good. Note that your derivation would work for any angle, not just for 45 degrees. If ##\theta_2## is the angle you are looking for, how would you write your equation with 45 degrees replaced by ##\theta_2##?
 
  • #3
Well, I could substitute ##\Theta_2##:
##mg=q \cdot E \cdot \tan(\Theta_2)##
##\tan(\Theta_{2})=\frac{mg}{q \cdot E}##
##\Theta_2=\arctan \frac{mg}{q \cdot E}##

What do I do now? Do I just say that ##\Theta_2## is 60% of 45 degrees which is 27 degrees?
 
  • #4
steroidjunkie said:
Well, I could substitute ##\Theta_2##:
##mg=q \cdot E \cdot \tan(\Theta_2)##
##\tan(\Theta_{2})=\frac{mg}{q \cdot E}##
##\Theta_2=\arctan \frac{mg}{q \cdot E}##
OK. Here ##q## is the charge corresponding to ##\theta_2##.

What do I do now? Do I just say that ##\Theta_2## is 60% of 45 degrees which is 27 degrees?
No. You have two equations, one for the initial charge ##q_1## and one for the final charge ##q_2##. Combine them to find ##\theta_2##.
 
  • #5
OK, than:

##\tan 45=\frac{m \cdot g}{q_1 \cdot E}##
##\tan \Theta_2=\frac{m \cdot g}{q_2 \cdot E}##

##q_1=q, q_2=q_1-40\%=q-0.4q=0.6q####\tan 45=\frac{m \cdot g}{q \cdot E}##
##\tan \Theta_2=\frac{m \cdot g}{0.6 \cdot q \cdot E}##

##q \cdot \tan 45=\frac{m \cdot g}{E}##
##0.6 q \cdot \tan \Theta_2=\frac{m \cdot g}{E}##

I equate the two equations:
##0.6 q \cdot \tan \Theta_2=q \cdot \tan 45##

##\tan \Theta_2=\frac{\tan 45}{0.6}##

##\tan \Theta_2=\frac{1}{0.6}##

##\tan \Theta_2=\frac{5}{3}##

##\Theta_2=59^{\circ} 04'##

I'm sorry for writing minutes the way I did. I couldn't find a proper code.
 
  • #6
OK. Good work.

My calculator gives the answer as a decimal number of 59.04o.
.04 degree is about 2 minutes.

Considering that the 45 degrees is given to 2 significant figures, it would probably be best to express the answer as 59o or maybe 59.0o.
 
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  • #7
You're right. It's 2 minutes, and not 4 minutes. Thank you very much for helping me.
 

What is a metal sphere on a thread in a horizontal electric field?

A metal sphere on a thread in a horizontal electric field is an object used to demonstrate the effects of an electric field on a conductive material.

How does a metal sphere on a thread behave in an electric field?

The metal sphere on a thread will experience a force in the direction of the electric field. This force will cause the sphere to move in the direction of the field.

What factors affect the behavior of a metal sphere on a thread in an electric field?

The behavior of a metal sphere on a thread in an electric field is affected by the strength of the electric field, the charge on the sphere, and the distance between the sphere and the source of the electric field.

Why is a metal sphere used in this experiment instead of another material?

A metal sphere is used because it is a good conductor of electricity. This allows for a stronger and more noticeable effect in the presence of an electric field.

What is the purpose of using a horizontal electric field in this experiment?

A horizontal electric field is used to demonstrate the effects of a constant electric field on the motion of the metal sphere. It allows for a clear and consistent demonstration of the principles of electricity and conductive materials.

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