I was able to come up with the answers through trial and error, and the answer for the charge +q makes sense, but I don't understand why the answer for the -q is correct. Please help.
A positive point charge +q1, a second point charge q2 that may be positive or negative, and a spot labeled P, are all on the x axis in a straight line. The distance d between the two charges is the same as the distance between q1 and P. With q2 present, the magnitude of the net electric field at P is twice what it is when q1 is present alone. Given that q1 = +2.86 µC, determine the magnitude |q2| when q2 is the following.
a) q2 is positive.
b) q2 is negative.
The correct answer for part a is that q2 = 4p.
The correct answer for b is that q2 = 12p.
I don't see why, especially in part b. Can anyone explain?
To understand why the answer for part (b) is q2 = 12p, let's break down the problem and analyze the given information.
First, we know that when only the positive charge +q1 is present, the magnitude of the net electric field at point P is Ex1.
When we introduce a second charge q2, we are interested in finding the magnitudes of q2 that will result in double the electric field at point P compared to when only q1 is present. Let's denote this as 2Ex1.
The net electric field at point P, due to the combination of q1 and q2, will depend on the distances between them.
Since the distances between q1 and P, and between q1 and q2 are equal (given in the problem as d), we can use a symmetry argument. This means that if we find the correct magnitude of q2 such that the electric field at point P is double with respect to the case of only q1, this solution will work regardless of the placement of q2 (left or right of q1).
Now, let's consider part (b) where q2 is negative.
When q2 is negative, it means it has an opposite charge compared to q1. In other words, q2 and q1 create opposite electric fields due to their opposite charges.
Since the electric fields created by q1 and q2 are in opposite directions, they add together vectorially. And when q2 is negative, its electric field vector points towards the left, whereas the electric field vector due to q1 points towards the right.
To achieve a net electric field at point P that is double with respect to the case of only q1, the electric field created by q2 has to be equal in magnitude to that of q1 but in the opposite direction.
This implies that the magnitude of the electric field due to q2 should be equal to Ex1.
Therefore, for q2 to be equal to 2Ex1 in magnitude, and for q2 to create an electric field that cancels the electric field of q1, the magnitude of q2 must be equal to 2q1.
Substituting the value of q1 (2.86 µC) into the equation, q2 = 2q1 = 2(2.86 µC) = 5.72 µC.
But wait, we're not done yet!
The problem asks for the magnitude of q2, so the answer should be a positive value. Thus, the magnitude of q2 when q2 is negative should be |q2| = 5.72 µC.
Therefore, the correct answer for part (b) is |q2| = 5.72 µC, not 12p (which seems to be a typo in the question).
To summarize:
a) When q2 is positive, |q2| = 4p.
b) When q2 is negative, |q2| = 5.72 µC.