# Electric field of a Ring

• tuggler
In summary: Different methods can lead to different results, but that is okay.Different methods can lead to different results, but that is okay.Yes, that is definitely true.Different methods can sometimes give you different results, but that is okay.

## Homework Statement

I am suppose to find an expression for the electric field of a ring.

## Homework Equations

$$E =\frac{Kq}{r^2}$$

## The Attempt at a Solution

I calculated my results and I reached up to this:

$$\frac{Kx\Delta q}{(R^2 + x^2)}^{3/2}$$ where R = radius, x = distance, K = constant, q = charge.

And then I looked at my book and noticed they integrated with respect to $$\Delta q$$ which got me confused because $$\Delta q$$ is not a geometric property. And the final expression the book got was $$\frac{Kx q}{(R^2 + x^2)}^{3/2}$$.

How come they can integrate with respect to $$\Delta q$$?

You can integrate over the total charge in the same way you can integrate over the length of the ring, or over the total mass to calculate the moment of inertia. The integration variable does not have to be a length, it can be nearly everything.

Integration is just summation of infinitely small elements of something. Here you are adding up all the Electric field due to all the Δq of the ring at a point along the axis at a distance x. So according to your equation

$\sum \Delta E =\sum \frac{K x\Delta q}{(x^2+y^2)^{3/2}}$
becomes,
$\int \ dE =\int \frac{K x dq}{(x^2+y^2)^{3/2}}$
when Δq→0

But when I was measuring the electric field of a line of charge I got the expression $$\sum \frac{d \Delta Q}{(y_1^2 + d^2)^{3/2}}$$ but with that expression I can't integrate over $$\Delta Q$$ because its not a geometric quantity so I had to replace $$\Delta Q$$ with $$\Delta Q = Q/L \Delta y.$$ I don't understand why we had to change it with the field of a line but not with a disk?

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Integration variables don't have to be geometric quantities.

I don't understand why we had to change it with the field of a line but not with a disk?
You don't have to in both cases.

I am not understanding, then why did my book try and trick me?

They did those two problems different with the same reasoning I mentioned above. And they did the same thing with an electric field of a disk as they did with the line of charge by replacing $$\Delta Q$$ with the density over the surface area $$2r\pi dr$$.

tuggler said:
I am not understanding, then why did my book try and trick me?
I don't think it did.
If there are multiple ways to solve a problem, you have to choose one. There is no "right" or "wrong" choice.

## 1. What is the formula for calculating the electric field of a ring?

The formula for calculating the electric field of a ring is E = kQz/(z^2 + R^2)^(3/2), where k is the Coulomb's constant, Q is the electric charge of the ring, z is the distance from the center of the ring to the point where the electric field is being calculated, and R is the radius of the ring.

## 2. How does the electric field of a ring vary with distance from the center?

The electric field of a ring decreases with distance from the center, following an inverse square relationship. This means that as the distance from the center increases, the electric field strength decreases at a faster rate.

## 3. Can the electric field of a ring be negative?

Yes, the electric field of a ring can be negative. This occurs when the electric charge of the ring is negative, resulting in an electric field that points towards the center of the ring.

## 4. What is the direction of the electric field at the center of a ring?

At the center of a ring, the electric field is zero. This is because the electric field contributions from all points on the ring cancel each other out.

## 5. How does the electric field of a ring compare to that of a point charge?

The electric field of a ring is different from that of a point charge in that it is not uniform. The electric field of a point charge is the same at all points in space, while the electric field of a ring varies with distance from the center. Additionally, the electric field of a ring is always perpendicular to the ring, while the electric field of a point charge can have any direction.