A 10 µC charge is at the origin. A -5 µC charge is on the x-axis 10 cm to the right of the origin.

Question

A 10 µC charge is at the origin. A -5 µC charge is on the x-axis 10 cm to the right of the origin. At what point other than at infinity can a 1 µC charge be placed so that there will be no net electrostatic force on it?

 

At one point on the x-axis to the left of positive charge

 

At no point

 

At some point off the x-axis either above or below

 

At one point on the x-axis to the right of the negative charge

 

At one point between the two charges

Answer 1

Hi Brooke!

 

Thanks for reposting the question with the multiple choice answers.  I was going to work up a way to actually calculate the position.  But now we can look at the answer more qualitatively (without calculation).

 

There are a couple main ways to approach this problem.  I am going to describe it in terms of electric fields.  This is because the expression for electrostatic force F_E can be related to the net electric field, E, in a region by the defining equation:

 

F_E = qE

 

where q is some charge (in this case, the 1 μC charge) placed in the electric field.  As long as q is not zero (and q=0 would be a trivial case, anyway), the only way F_E = 0 at some point is if the net electric field E = 0 at that point.  This is regardless of the magnitude and sign of the charge you put at that point.

 

So where can the net electric field E = 0 in this case?  Each charge produces its own electric field, that either points toward the charge (for negative charges) or away from the charge (for positive charges) at every point in space around the charge.  The magnitude of the electric field, |E|, generated by a point charge at a distance r from the charge is given by:

 

|E| = kq/r²

 

Each of the two charges listed in the problem creates its own electric field at every point in space around it.  The net electric field at any point is the vector sum of the field from each charge at that point.  To have those two fields cancel out, two conditions are needed:

 

1.  The electric fields from each charge have to point in opposite directions

2.  The electric fields from each charge have to have equal magnitude

 

We can look at the two conditions separately.  I often look at the first one first, since that one (about directions) can often eliminate a lot of possible answers by inspection.

 

For example, if you were to draw a diagram with the two charges on the x-axis as specified, and then picked any point off the x-axis, we can see right away that such a point does not meet Condition 1.  This is because the electric field from the (+)10 μC charge will point from that position away from the 10 μC charge.  And the electric field from the -5 μC will point from that position toward the -5 μC charge.  For any position off the x-axis, these two electric field vectors will not even lie along the same line.  Hence, even if their magnitudes are equal, they cannot cancel out, because they cannot point in opposite directions.  So this eliminates the third answer above.

 

So let's look on the x - axis.  For positions on this line, the electric fields point along the same line, and thus can possibly point in opposite directions.  But where is this possible?  For a line with two charges, we can divide the line into three regions: (I) to the left of both charges, (II) between the charges, and (III) to the right of both charges.

 

Region (II)

Even though it is not given as a possible answer, let's look at between the charges first.  In this region, the electric field from the 10 μC points away from that (positive) charge, to the right; and the electric field from the -5 μC charge points toward that positive charge; also, to the right.  So the electric fields point the same way, at any point between the charges.  This violates Condition 1, so the net electric field cannot be zero between the charges.

 

Region (I)

To the left of both charges, the 10 μC charge's electric field still points away from the charge, but that is now to the left.  We are still on the same side of the -5 μC charge, so that charge's electric field points to the right.  Thus, the two fields point in opposite directions in Region (I), fulfilling Condition 1 above.  So Region (I) could contain a point, or points, where E_net = 0.

 

Region (III)

To the right of both charges, the 10 μC charge's electric field points away from the charge, which is to the right.  And the -5 μC charge's electric field points toward that charge, which is to the left.  So the electric fields do point in opposite directions in Region (III), fulfilling condition 1 and making it another possibility for including points where E_net=0.

 

So Condition 1 has reduced the possibilities for existing E_net = 0 positions to Region (I) (to the left of both charges) and Region (III) to the right of both charges.  Now let's bring in Condition 2.

 

For an analogy suggesting where the two electric field magnitudes might be equal, think of two light bulbs, one of which shines brighter than the other.  If you wanted to position yourself so that the two bulbs appeared, to you, to have equal brightness, what would you have to do?  Without doing any explicit calculations, we can say that you would have to place yourself closer to the dimmer bulb.  Then its apparent brightness to you could be the same as the brighter-shining bulb that is farther away.  If you line of the two bulbs and look at them along that line, there should be one specific point where you are the right distance from both bulbs for their apparent brightnesses, to you, to be equal.

 

For an analogous reason, to get the two magnitudes of electric fields from two unequal electric charges to be the same (regardless of the sign of the charge; that only matters regarding direction, not magnitude), you have to place yourself closer to the smaller-magnitude charge.  So, to have any hope of fulfilling Condition 2, we have to be at one specific point, closer to the smaller-magnitude charge.

 

So, our two conditions mean that, to be at a position where E_net = 0, we have to be:

 

1.  either in Region (I) (to the left of both charges) or Region (III) (to the right of both charges); and

2.  be at one specific point that is closer to the smaller charge

 

These two conditions, together, indicate one of the answers above.  If you want to talk about any part of this in more detail, or check an answer, just let me know.

 

I hope this helps!