Understanding Net Electric Fields: Explaining Charges and Magnitudes

[smunger81]

I'm pretty sure I'm making this question a lot harder than it is...

The question is this - "Three point charges are fixed to the corners of a square, one to a corner, in such a way that the net electric field at the empty corner is zero. Do these charges all have the same sign or the same magnitude but perhaps different signs?"

My answer is the charges would have the same sign and different magnitudes because when calculating net electric fields the sign of the charge is not taken into account...is this right?

Any advice would be fabulous! Thank you!

 

[Doc Al]

Your answer and reason are incorrect. The sign of the charges must be taken into account in finding the electric field.

 

[smunger81]

Wow...then I am really confused. Could you shed some light on that for me?

(Thanks for your response!)

 

[Doc Al]

Here's a hint: Somehow the fields from each of the three charges must "cancel out" at the fourth corner. So, what does that tell you about the signs of the charges?

 

[smunger81]

Ok...so in order for the signs to cancel each other out you need at least one sign that was negative...since the field at the empty corner (q4) of the square is the vector sum of the fields due to q1, q2, and q3. And since each of the charges is equidistant from each other and the signs are different, the magnitudes of the electric fields would be equal. Am I getting closer?!

 

[robphy]

are the three charges "equidistant" from the empty corner of the square?

 

[Doc Al]

Closer. Note that what counts is the distance from the charges to the fourth corner. Note that q1 and q3 are the same distance away from the empty corner, but q2 is on a diagonal and thus further away.

Since q1 and q3 are on opposite sides of the diagonal, what can you say about their charges? Are they the same? Different?

 

[smunger81]

They would be opposite in order to cancel each other out...

 

[smunger81]

So if those two charges cancel each other out...you would have a zero electric field at one corner and a ? charge at the other...

 

[Doc Al]

They would be opposite in order to cancel each other out...

Draw yourself a picture showing the fields produced by the three charges at that fourth corner. Remember that the field from a positive charge points away from the charge.

 

[smunger81]

If both charges on opposite sides of the diagonal were positive you would have one charge pointing in the -x, -y direction and one pointing in the +x, +y direction. So those two would cancel each other out. The other charge (on the diagonal) is a length = sq.root(a^2+b^2) away from the empty corner, according to pythagorean's theorem. But since it is a square, all of the sides are equal length of d so a and b would be equal. Thus the above distance equation would = sq.root(d^4). So how would a negative charge at this corner allow the empty corner to have an electric field of 0?

 

[Doc Al]

To make things easier to describe, let's imagine the empty corner is at the origin and the opposite diagonal (where q2 is) is somewhere on the y-axis. Thus q1 (on the left side of the y-axis) and q3 (on the right side) have the same y-coordinate, but opposite x-coordinates. Got it?

If both charges on opposite sides of the diagonal were positive you would have one charge pointing in the -x, -y direction and one pointing in the +x, +y direction.

Careful. Using my coordinate system the field from q1 will have components along the +x and -y directions; the field from q3 will have components along the -x and -y directions. So if q1 and q3 are both positive and equal, then the x-components would cancel, but not the y-components--those will add.

So those two would cancel each other out.

See my comment above. The net field from q1 and q3 will point in the -y direction.

The other charge (on the diagonal) is a length = sq.root(a^2+b^2) away from the empty corner, according to pythagorean's theorem. But since it is a square, all of the sides are equal length of d so a and b would be equal. Thus the above distance equation would = sq.root(d^4). So how would a negative charge at this corner allow the empty corner to have an electric field of 0?

If the sides are length d, the diagonal is length $\sqrt{2} \ d$. Since the field from q1 and q3 points down (-y direction), to cancel that field you need to add a field pointing up (+y direction). A negative q2 will do that, but it needs to be the right size charge to cancel the other fields exactly.

 

[smunger81]

Wow. Your way is waay easier to understand! Like I said at the beginning, I was making it a lot harder than it really was. Using a square diagram to draw the components I was completely screwing up which direction they were going. As I reread my last post and look at my diagrm...I don't know what I was thinking. It makes sense now when looking at it from the perspective that the empty corner is at the origin. Thank you for your assistance in working through this problem with me...I know I didn't do a very good job but I tried my hardest and I appreciate your not giving up on the question. Thanks again for your help.