Finding the Maximum Electric Field of a Charged Ring

[05holtel]

Homework Statement

Consider a charged ring of radius 20.6 cm and total charge 12 nC.

We are interested in the electric field a perpendicular distance z away from the center of the ring.

At what distance from the center of the ring does the electric field become maximum?

Hint: The field for a ring of charge is:

Ering = (1/4 pi eo) (zQ/(z^2+R^2)^(3/2))

Homework Equations

The Attempt at a Solution

When x = 0, the fields due to segments of the ring cancel out. As x -> infinity, the field falls with 1/x^2 behavior, so there has to be a maximum E for some x.

When adding up the fields due to each arc segment, you only have to add the x-components (along the axis) because the others will cancel out.

Here is what I get for E as a function of x:

E (x) = [k*Q /(x^2 + r^2)]*[x/sqrt(x^2+r^2)]
The second term in brackets is the cosine of the angle that defines the component in the x direction.

That function must be differentiated to find where the field is a maximum.


1) is this right so far
2) if so how do I differentiate this

 

[pgardn]

Homework Statement

Consider a charged ring of radius 20.6 cm and total charge 12 nC.

We are interested in the electric field a perpendicular distance z away from the center of the ring.

At what distance from the center of the ring does the electric field become maximum?

Hint: The field for a ring of charge is:

Ering = (1/4 pi eo) (zQ/(z^2+R^2)^(3/2))

Homework Equations

The Attempt at a Solution

When x = 0, the fields due to segments of the ring cancel out. As x -> infinity, the field falls with 1/x^2 behavior, so there has to be a maximum E for some x.

When adding up the fields due to each arc segment, you only have to add the x-components (along the axis) because the others will cancel out.

Here is what I get for E as a function of x:

E (x) = [k*Q /(x^2 + r^2)]*[x/sqrt(x^2+r^2)]
The second term in brackets is the cosine of the angle that defines the component in the x direction.

That function must be differentiated to find where the field is a maximum.


1) is this right so far
2) if so how do I differentiate this

Well if the field starts off at zero at x= 0 then increases, and then dies off again, then I think we would look at where dE/dx = 0 as this is where the slope of an E v. x graph would be maximum.

In the equation given, they give you the e-field function of this ring, with Z being your X? Or is Z supposed to be the radius of the ring, or is R the radius of the ring and not the distance from a specific segment of the ring of charge to the point of interest?

 

[pgardn]

In others words, r is usually the distance from some charge to some place you are looking for the E field. In the equation you were given, is r the radius of the charged ring, or the distance from the charge to the place of interest..?