#### Where is the Field Equal to Zero?

Two charges, +3Q and -Q, are separated by 4 cm. Is there a point along the line passing through them (and a finite distance from the charges) where the net electric field is zero? If so, where?

More specifically, is the field equal to zero at some point in one of these three regions: to the left of both charges (Region I), in between both charges (Region II), and/or to the right of both charges (Region III)?

The field is zero at a point in:

1. Region I
2. Region II
3. Region III
4. two of the above
5. all of the above

In Region I, to the left of both charges, the fields from the two charges are in opposite directions, which is what we need for them to cancel. However, in region I we are always closer to the larger charge. Both the smaller r and the larger q act to make the field from the positive charge significantly larger than that from the negative charge, so they can't cancel one another.

In Region II, between the charges, both vectors point in the same direction so there is no possibility of cancelling out.

In Region III, the fields again point in opposite directions and there is a point where their magnitudes are the same. It is at this point where the net electric field is zero.

What happens at this point? Because F = qE, if there is no electric field at a point then a test charge placed at that point would feel no force.

How can we calculate where the point is? If the point is a distance x from the +3Q charge, then it is x-4 away from the -Q charge. If we define right as positive, we can write this as:

k (3Q / x2) - k (Q / (x - 4)2) = 0

where the minus sign in front of the second term is not the one associated with the charge but the one associated with the direction of the field from the charge.

The k's and Q's cancel. Re-arranging gives:

3 / x2 = 1 / (x - 4)2

Cross multiplying and expanding the brackets:

3(x2 - 8x + 16) = x2

x2 - 12x + 24 = 0

Solving this using the quadratic equation gives two answers: x = 2.54 cm and x = 9.46 cm. Which answer should we keep?

1. 2.54 cm
2. 9.46 cm