Placing charges with coulomb's law problem

[kirby2]

Problem: a charge +Q is located at the origin and a second charge, +4Q is at a distance d on the x-axis. where should a third charge, q, be placed, and what should be its sign and magnitude, so that all three charges will be in equilibrium.

attempt:

I don't know how to solve it fully, but my initial thought is to put the third charge in between the other two and give it a negative sign. The two other charges repel each other so i thought a negative one in the middle would put them in equilibrium. is this right?

 

[SammyS]

Problem: a charge +Q is located at the origin and a second charge, +4Q is at a distance d on the x-axis. where should a third charge, q, be placed, and what should be its sign and magnitude, so that all three charges will be in equilibrium.

attempt:

I don't know how to solve it fully, but my initial thought is to put the third charge in between the other two and give it a negative sign. The two other charges repel each other so i thought a negative one in the middle would put them in equilibrium. is this right?

You are right about the sign of the third charge.

Considering, for a moment, only the positive charges:

1. Is there any location along the x-axis where the electric field is zero?

2. What is the magnitude of the force that the two positive charges exert on each other?​

Answering those should give you a good start.

 

[kirby2]

am i correct by saying that there is no point on the x-axis where the electric field is zero

 

[SammyS]

am i correct by saying that there is no point on the x-axis where the electric field is zero

No.

For any location between x = 0 and x = d, The direction of the electric field due to the +Q charge is in the opposite direction of the electric field due to the +4Q charge . Therefore, there is some location between x = 0 and x = d at which the electric field due to those two charges is zero. At that location, what is true of the magnitude of the field due to +Q compared to the magnitude of the field due to +4Q ?

In my experience, by the time you encounter a problem like the one in this thread, you would have already done some problems with two charges located on the x-axis and you are to find the location at which the electric field is zero.
.

 

[kirby2]

i think i got it. it should be placed right d/3 from origin. sign negative. and charge of 7.11 x 10^(-20).

 

[SammyS]

i think i got it. it should be placed right d/3 from origin. sign negative. and charge of 7.11 x 10^(-20).

That's the correct location.

You haven't given a specific value for the charge Q. How can you give a specific numerical value for the 3rd charge, q ?

Use the second hint I gave you.

 

[kirby2]

i assumed Q to be 1.6 x 10^(-19)

 

[SammyS]

i assumed Q to be 1.6 x 10^(-19)

Any reason for that assumption?

If you assume that, then the third charge, q, will be a fraction af one electron charge.

 

[kirby2]

the force that the two charges exert on each other is (k)((4Q^2)/d^2). but i don't know how that helps

 

[SammyS]

the force that the two charges exert on each other is (k)((4Q^2)/d^2). but i don't know how that helps

That is a force of repulsion.

The middle charge, q, must exert a force equal to that. As you have concluded previously, charge, q, must have a sign opposite to that of Q & 4Q, so that it attracts both of them.

What is the (magnitude of the) force exerted on each other by the charge, Q, at the origin and the charge, q, at x = d/3 ?

Set that force equal to $\displaystyle \frac{4k\,Q^2}{d^2}\,.$

 

[kirby2]

OK. i think its -3Q.

 

[SammyS]

OK. i think its -3Q.

Magnitude of the force between Q & 4Q:

$\displaystyle \frac{4k\,Q^2}{d^2}\,.$​

Magnitude of the force between Q & q: assuming q is at x=d/3 and q = -3Q:

$\displaystyle \frac{4k\,(3Q)\cdot Q}{(d/3)^2}=27\frac{4k\,(Q^2)}{d^2}\,.$​