Seperation between two charged ball

In summary, the two masses are separated by a distance x if the potential energy between the masses is zero, but the energy from the electric field is at its greatest.
  • #1
9
0

Homework Statement


Two identical insulating balls of mass m hang from massless strings of length l and carry identical electric charges, q. you may assume that the angle of separation θ is so small that tanθ ≅sinθ≅θ.
1.png

What is the separation distance between the two masses x?

The Attempt at a Solution


I made two attempts at a solution, though it has been so long since I worked a problem like this that I don't know if either attempt is correct (I do know that both methods give different values for x, so at least one is incorrect).

1st using energy conservation:
2.png


I believe this method gives the incorrect answer as there is some initial energy between the two particles I have not accounted for, when the potential energy is zero but the energy from the E-field is at its greatest. Because I do not know the diameter of the balls, or their starting position, I assume this method is a bad one?

2nd method using statics:
3.png

Based off the free body diagram...
4.png

thus
5.png

so then
6.png

where
7.png

as such
8.png

and finally
9.png

Is this correct? Can someone please point me in the right direction?
 

Attachments

  • 1.png
    1.png
    7.1 KB · Views: 547
  • 2.png
    2.png
    1.6 KB · Views: 228
  • 3.png
    3.png
    3.7 KB · Views: 227
  • 4.png
    4.png
    1.1 KB · Views: 226
  • 5.png
    5.png
    627 bytes · Views: 217
  • 6.png
    6.png
    908 bytes · Views: 232
  • 7.png
    7.png
    740 bytes · Views: 236
  • 8.png
    8.png
    1.1 KB · Views: 226
  • 9.png
    9.png
    923 bytes · Views: 221
Last edited:
Physics news on Phys.org
  • #2
Lady M said:
I believe this method gives the incorrect answer as there is some initial energy between the two particles I have not accounted for, when the potential energy is zero but the energy from the E-field is at its greatest.
Right. You don't know the total energy in the system, and the particles being at the same place is an unphysical situation.
Lady M said:
2nd method using statics:
Shouldn't there be some equations in this part?
 
  • #3
mfb said:
Right. You don't know the total energy in the system, and the particles being at the same place is an unphysical situation.Shouldn't there be some equations in this part?
My images did not upload. Give me a second... Sorry.

Edit: I've fixed it now. Not sure why it didn't work the first time...
 
  • #4
Was there any reson for you to doubt your second method?
 
  • #5
Chandra Prayaga said:
Was there any reson for you to doubt your second method?

The problem mentioned that I should be aware of the small angle approximation, which I did not use. Also, I was uncertain if I would have to account for the second ball in this problem. I feel like I had already done so, but I wanted to be certain of that.
 
  • #6
You expressed x as function of the angle (and other constants). The answer should not depend on the unknown angle. If you could use the angle in the answer, simple geometry would be much faster.
 
  • #7
mfb said:
You expressed x as function of the angle (and other constants). The answer should not depend on the unknown angle. If you could use the angle in the answer, simple geometry would be much faster.

Yes, you could just take the sin of that angle, and multiply it by twice the length of the string.

So is there a better way to do this? Is there a way to do this without knowing the final angle of separation or the distances between the particles?
 
  • #8
Notice that the final result you got contains both x and θ. You can eliminate one of them simply in the small angle approximation. You can use:

tanθ≈sinθ=x/l

and then solve for x in terms of just the mass and charge of each ball.
 
  • #9
Chandra Prayaga said:
Notice that the final result you got contains both x and θ. You can eliminate one of them simply in the small angle approximation. You can use:

tanθ≈sinθ=x/l

and then solve for x in terms of just the mass and charge of each ball.
Ah, so then
10.png
would be the final answer in terms of everything we "know"?
 

Attachments

  • 10.png
    10.png
    783 bytes · Views: 530
  • #11
Chandra Prayaga said:
Notice that the final result you got contains both x and θ. You can eliminate one of them simply in the small angle approximation. You can use:

tanθ≈sinθ=x/l
No, sinθ=x/(2l)
 
  • #12
Lady M said:
Ah, so then View attachment 229589 would be the final answer in terms of everything we "know"?
Not quite, because
Chandra Prayaga said:
sinθ=x/l
Should be sinθ=x/(2l)

... beaten to it by ehild.
 
  • #13
Oops, sorry. As seen in the diagram, ehild above is right. I overlooked that.
 
  • #14
ehild said:
No, sinθ=x/(2l)
Yes, you are correct, as we want twice the length of x (the sin part of two triangles). As such the final solution is
10.png
.

Thanks to everyone who has helped me with this problem.
 

Attachments

  • 10.png
    10.png
    844 bytes · Views: 192
  • 11.png
    11.png
    589 bytes · Views: 247

What is the concept of separation between two charged balls?

The separation between two charged balls refers to the distance between the centers of the two balls. It is an important factor in determining the strength of the electric force between the two balls.

How is the separation between two charged balls related to the electric force between them?

The electric force between two charged balls is directly proportional to the product of their charges and inversely proportional to the square of the separation between them. This means that as the separation decreases, the electric force increases and vice versa.

What happens to the electric force between two charged balls if the separation is doubled?

If the separation between two charged balls is doubled, the electric force between them decreases by a factor of four. This is because the force is inversely proportional to the square of the separation.

Is there a minimum or maximum separation between two charged balls?

There is no specific minimum or maximum separation between two charged balls. However, as the separation approaches infinity, the electric force between them becomes negligible.

How does the type of charge on the two balls affect the separation between them?

The type of charge on the two balls (positive or negative) does not affect the separation between them. However, it does affect the direction of the electric force between them, with like charges repelling each other and opposite charges attracting each other.

Suggested for: Seperation between two charged ball

Back
Top