Related Rates involving circular ring

[Painguy]

Homework Statement

A circular ring of wire of radius r₀ lies in a plane perpendicular to the x-axis and is centered at the origin. The ring has a positive electric charge spread uniformly over it. The electric field in the x-axis direction, E, at the point given by

E=kx/((x^2 +r₀^2)^(3/2)) for k>0

at what point on the x-axis is greatest? least?

Homework Equations

The Attempt at a Solution

so the only thing i could really think of to do is take the derivative. the circle itself isn't changing so I assumed r0 is a constant as well as k.

$E'=(k(x^2 + r₀^2)^(3/2) - 3kx^2√(x^2 +r₀^2))/((x^2 +r₀^2)^(3/2))$after this i find the critical points

0=(k(x^2 + r₀^2)^(3/2) - 3kx^2√(x^2 +r₀^2))/((x^2 +r₀^2)^(3/2))

0=(x^2 +r0^2)^(3/2) -3x^2√(x^2+r0^2)

im not sure what do here.

I feel like I should have solved for r0 in terms of x, but I am not sure.

 

[SammyS]

Homework Statement

A circular ring of wire of radius r₀ lies in a plane perpendicular to the x-axis and is centered at the origin. The ring has a positive electric charge spread uniformly over it. The electric field in the x-axis direction, E, at the point given by

E=kx/((x^2 +r₀^2)^(3/2)) for k>0

at what point on the x-axis is greatest? least?

Homework Equations

The Attempt at a Solution

so the only thing i could really think of to do is take the derivative. the circle itself isn't changing so I assumed r0 is a constant as well as k.

$E'=(k(x^2 + r₀^2)^(3/2) - 3kx^2√(x^2 +r₀^2))/((x^2 +r₀^2)^(3/2))$

after this i find the critical points

0=(k(x^2 + r₀^2)^(3/2) - 3kx^2√(x^2 +r₀^2))/((x^2 +r₀^2)^(3/2))

0=(x^2 +r0^2)^(3/2) -3x^2√(x^2+r0^2)

im not sure what do here.

I feel like I should have solved for r0 in terms of x, but I am not sure.

The problem says to solve for x.

 

[Mark44]

Homework Statement

A circular ring of wire of radius r₀ lies in a plane perpendicular to the x-axis and is centered at the origin. The ring has a positive electric charge spread uniformly over it. The electric field in the x-axis direction, E, at the point given by

E=kx/((x^2 +r₀^2)^(3/2)) for k>0

at what point on the x-axis is greatest? least?

Some information is missing here. I'm pretty sure you are asked where E is the greatest and least.

The Attempt at a Solution

so the only thing i could really think of to do is take the derivative. the circle itself isn't changing so I assumed r0 is a constant as well as k.

$E'=(k(x^2 + r₀^2)^(3/2) - 3kx^2√(x^2 +r₀^2))/((x^2 +r₀^2)^(3/2))$


after this i find the critical points

0=(k(x^2 + r₀^2)^(3/2) - 3kx^2√(x^2 +r₀^2))/((x^2 +r₀^2)^(3/2))

Leaving the numerator as a difference isn't much help. The usual thing to do when you use the quotient rule is to find the greatest common factor of the terms in the numerator. It's also better to leave both of the parts that involve x² + r₀² in their exponent form, rather than switch to the radical form for one, as you have done.

Once you find and pull out the greatest common factor of the terms in the numerator, the numerator will be a product of factors, and it will be easy to find the values of x for which E'(x) = 0.

0=(x^2 +r0^2)^(3/2) -3x^2√(x^2+r0^2)

im not sure what do here.

I feel like I should have solved for r0 in terms of x, but I am not sure.