Two small identical conducting spheres are separated by a distance L horizontall

Question

Two small identical conducting spheres are separated by a distance L horizontally that is much larger than their radii. The leftmost of the two spheres experiences a force F directed to the right due to the presence of the second sphere. The two spheres are brought into contact and 5x10^12 electrons are transferred from one sphere to the other (the charge of an electron is 1.6x10^-19 Coulombs). The spheres are then separated by a distance L again and the leftmost sphere experiences a force of magnitude 4/5 F.

A) Describe the initial charge of the two spheres. Explain your reasoning

B) What is the direction of the force on the leftmost sphere after the objects have been in contact? Explain.

C) Write an expression relating the magnitude of the force before the spheres touch to that after they touch.


D) Imagine that the two charged objects had charges of Q1 and Q2. When they are touched they transfer an amount of charge ?Q. Write an expression involving Q1and Q2 (total, two expressions) in terms of the total charge of the system(1) and the amount of charge transferred (2).

E) Using the expression for E, find equations for Q1and Q2 in terms of the total charge and ?Q

F) Find the initial charge on each of the spheres before they were touched.

Explanation / Answer

A) Describe the initial charge of the two spheres. Explain your reasoning
Initialy, they are attracted to each other. So two spheres have different sign charges.
B) What is the direction of the force on the leftmost sphere after the objects have been in contact? Explain.
to the left. Because when the two spheres are in contacted, the amount of charges as well as its sign of the two spheres are exactly the same. So they push each others.
C) Write an expression relating the magnitude of the force before the spheres touch to that after they touch.
F_initial = -5/4 * F_after
minus because they have different propertie( one is atractive while one is repulsive)
D) Imagine that the two charged objects had charges of Q1 and Q2. When they are touched they transfer an amount of charge ?Q. Write an expression involving Q1and Q2 (total, two expressions) in terms of the total charge of the system(1) and the amount of charge transferred (2).

total amount of charge. Q1+Q2
The amount of charge after the touch is the same for both spheres.

Q'=(Q1+Q2)/2.

transfered amount of charge.(from Q2 to Q1)

Q2-Q'=(Q2-Q1)/2
E) Using the expression for E, find equations for Q1and Q2 in terms of the total charge and ?Q
We have ?Q=(Q2-Q1)/2 and total charge = Q2+Q1.

so 2Q2=2?Q+Q_total

so Q2=?Q+Q_total/2. and Q1=Q_total/2-?Q
F) Find the initial charge on each of the spheres before they were touched.

We have F_initial is propose to Q1*Q2 and F_after is propotional to final charge ^2=Q_total^2/4.

so Q1*Q2=-5/4 *(Q1+Q2)^2/4=-5/16(Q1+Q2)^2.

Q1*Q2=(Q_total/2-?Q)*(Q_total/2+?Q)=Q_total^2/4-?Q^2.

so Q_total^2 *9/16=?Q^2.

so Q_total*3/4=?Q.

so Q_total = 0.75*?Q.

Q1=3/8*?Q-?Q=-5/8?Q=-5e-7(C).

Q2=3/8*?Q+?Q=11/8*?Q=11e-7(C)

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