Potential at a point due to two charges

[6Stang7]

I've been working on this problem and I cannot find out where I am making a mistake.

Homework Statement

Two charges, each with a value of +q, are placed a distance d apart on the x-axis. Find the potential at a point P a distance z above the x-axis on the z-axis

The Attempt at a Solution

The figure below shows the physics system:

The potential due to a charge is given by:

We can express r as:

We can see by symmetry that the x-components will cancel out, so the total potential will be twice of the z-component. The z-component of the potential is the cosine of the magnitude, where cosine is:

This gives us:

Putting everything together we get:

Therefore, the total potential at point P is:

Now, we know that the potential is related to the Electric Field by:

This means that the Electric Field at the same point is (I realize the forget the minus sign just now):

Which when solved give:

Now, using the same figure, but instead of solving for the potential, we solve for the Electric Field, we find:

The above uses the same line of thinking, and this gives a total value of the Electric Field to be:

The problem is that the book says that the potential I calculated above is wrong, and that the value should be:

When you calculate the value of the Electric Field from this, you get the same answer as the value I just calculated.

So, what did I do wrong in my calculation of the potential? The error seems to steam from the part where I calculated the z-component of one of the charges using cosine, but I can't see why that is incorrect.

 

[stevenb]

I've been working on this problem and I cannot find out where I am making a mistake.


So, what did I do wrong in my calculation of the potential? The error seems to steam from the part where I calculated the z-component of one of the charges using cosine, but I can't see why that is incorrect.

Just remember that potential is not a vector, but is a scalar. You don't add scalars vectorially.

 

[6Stang7]

So if one of the charges was negative instead of positive, the potential would be zero?

What confuses me is that if one charge was changed to negative, then there'd be a constant potential in the x-direction, but zero in the z-direction.

 

[stevenb]

So if one of the charges was negative instead of positive, the potential would be zero?

Yes, on the z axis it would be.

What confuses me is that if one charge was changed to negative, then there'd be a constant potential in the x-direction, but zero in the z-direction.

Remember, potential does not have a direction since it is a scalar and not a vector. There will be a force, or equivalently an electric field (which is force per unit charge) in the direction you say.

If potential is referenced to infinity, then the work done to bring a charge from infinity to any finite point on the z axis is clearly zero. Why, because you can chose the z axis itself as the path, and it's clear that the dot product of the electric field and the path is always zero. The x-component of electric field (and force) is orthogonal to the chosen path. The fact that electric fields are conservative tells you that any other path will yield the same value.