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Mock Exams AP Physics – 2 – FRQ Set 3

Question

Three charges are fixed at positions along the x-axis at positions -d, o, and +d. The charges at -d and +d are both negative, and the charge a o is positive.

(a) A positively charged object of mass m is placed on the x-axis between o and +d, close to the position x = o. If the three charges described above do not move as a result of this new charged object, describe the motion of the object after it is released as it moves in the region o < x < d. The charge at o has a magnitude of 2Q, while the other two
charges have a magnitude of Q.

(b) On the axes below, sketch the electric field along the x-axis in the vicinity of the charges. An electric field to the right should be graphed as positive and a field pointing left should be graphed as negative.

(c) Write a mathematical function, E(x), that gives the value of the electric field at any position along the x-axis for o < x < d. Give your answer in terms of Q, d, and fundamental constants. 

(d) In order to originally assemble the three charges on the x-axis, some work had to be done. Consider arranging the charges along the x-axis in the following manner: first, bring the +2Q charge to position x = o, then bring the -Q charge to x = +d, and finally, bring in the last charge. Bringing the +2Q charge to position o required no work. Bringing in the second charge required an amount of work W. Explain whether bringing in the third charge will require more work, less work, or an amount of work equal to W.

▶️Answer/Explanation

Ans:

(a) A positively charged object will be repelled from the positive charge at position o and attracted to the negative charge as position +d, so it will experience a force to the right during the entirety of its motion. The object will move from its release position directly towards the negative charge at position d with a velocity that increases during the entirety of the motion.
(b) The field will be to the right in the two regions -∞ < x < -d and o < x < d, so this is where the plot will be positive. The field magnitude will go towards infinity as the position approaches the charges as -d, o, and +d. The charge at o has a larger magnitude, so the positions where the charge has the smallest magnitude should be closer to the charges as +d and -d.

(c) In the region between o and d, the net charge will depend on the influence of each of the three charges on the x-axis.

\(E_{-d}(x)=\frac{kQ}{(d+x)^{2}}\) is directed to the left. 

\(E_{0}(x)=\frac{k(2Q)}{(x)^{2}}\) is directed to t e right. 

\(E_{+d}(x)=\frac{kQ}{(d-x)^{2}}\) is directed to the right. 

Thus, \(E(x)=\frac{kQ}{(d-x)^{2}}+\frac{k(2Q)}{(x)^{2}}- \frac{kQ}{(d+x)^{2}}\) directed to the right. 

(d) The work required to bring a charge to a position is equal to the electric potential at that position multiplied by the amount of charge that is moved. Once the first two charges have been assembled, the electric potential at position x = -d will be lower than when there was only a positive charge at x = o because the charge at x = d is negative. Therefore, the work to bring in the third charge will be less than W. 

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