Electrostatics problems
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The problem states:
Three small spheres with charge 2.00 mC are arranged in a line, with sphere 2 in the middle. Adjacent spheres are initially 8.0 cm apart. The spheres have masses m_1 = 20.0 g, m_2 = 85.0 g, and m_3 = 20.0 g. Their radii are much smaller than their seperation. The three spheres are released from rest.
It asks to:
1. Find the acceleration of sphere 1 just after it is released?
2. What is the speed of sphere 1 when the spheres are far apart?
3. What is the speed of sphere 2 when the speres are far apart?
4. What is the speed of sphere 3 when the spheres are far apart?
I understand that it is again a conservation of energy problem. I can calculate initial potential energy based on:
U = qV = qk(q_1/r_1 + q_2/r_1 + q_3/r_1)
I tried finding the electric field using this potential because of:
V = E*d with d = 0.08 m
Then I tried to find the Electrical force according to: F = q*E
Then I tried finding the acceleratoin given by: a = F/mass
I got 2.80 * 10^8 m/s^2 which is not the right answer.
Thank you!!!
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You can indeed solve the problem using conservation of energy. But you need to use the correct expression for the potential energy. If you have three particles with charges q_1, q_2 and q_3 at positions r_1, r_2 and r_3 respectively, then the potential energy is:
U(r_1, r_2, r_3) = k q_1 q_2/|r_1 - r_2| + k q_1 q_3/|r_1 - r_3| + k q_2 q_3/|r_2 - r_3|
Here |x| denotes the absolute value of x. Note that this formula is also valid in more than one dimension. Then the r_j are vectors and |x| denotes the length of the vector, so the |r_i - r_j| would then again be the distance from i to j.
The force that a particle experiences is minus the derivative w.r.t. its position coordinate. So, if you differentiate U w.r.t. r_1, you get minus the force exerted in the positive r_1 direction. To calculate the derivatives it is convenient to rewrite 1/|x| as sign(x)/x. Sign(x) = 1 if x> 0 and -1 if x < 0. If you calculate the derivative you find for the force exerted on charge 1 in the positive
F_1 = k q_1 q_2 sign(r_1 - r_2)/(r_1 - r_2)^2 + k q_1 q_3 sign(r1 - r3)/(r_1 - ...
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