Deriving formula for z coordinate and mass value.

[bobh.stein]

Homework Statement

Consider a very thin conducting ring of radius R which contains a total positive charge of +Q coulombs. (a) Derive a formula for the z coordinate which gives the maximum value for the magnitude of E ring (Z)? (b) Suppose that the charged ring is oriented horizontally, as shown in the diagram above right. Suppose that one wishes to levitate a ball with charge +Q coulombs above the charged ring. Derive a formula for the largest value for the mass M of a ball which can be levitated in this manner.

The diagram is shown here
http://www.chegg.com/homework-help/questions-and-answers/1-consider-conducting-ring-radius-r-contains-total-positive-charge-q-coulombs-derive-formu-q1409271

The Attempt at a Solution

I have no idea about how to start this question.

 

[cepheid]

Welcome to PF!

Homework Statement

Consider a very thin conducting ring of radius R which contains a total positive charge of +Q coulombs. (a) Derive a formula for the z coordinate which gives the maximum value for the magnitude of E ring (Z)? (b) Suppose that the charged ring is oriented horizontally, as shown in the diagram above right. Suppose that one wishes to levitate a ball with charge +Q coulombs above the charged ring. Derive a formula for the largest value for the mass M of a ball which can be levitated in this manner.

The diagram is shown here
http://www.chegg.com/homework-help/questions-and-answers/1-consider-conducting-ring-radius-r-contains-total-positive-charge-q-coulombs-derive-formu-q1409271

The Attempt at a Solution

I have no idea about how to start this question.

Well, it looks like you already have an expression for the electric field as a function of z. What would you need to do to find the maximum value of this function?

For levitating the ball: what condition must be satisfied by the two forces acting on the ball if it is to levitate?

 

[bobh.stein]

For 1 do you do the derivative of the function? and for part b is the charge the same as the gravitational force?

 

[cepheid]

For 1 do you do the derivative of the function?

Well, you took first-year calculus, did you not? How do you find the minima and maxima of functions?

and for part b is the charge the same as the gravitational force?

No, you can't equate charge with force, this is nonsensical. It's meaningless. Physical quantities on two sides of any equation have to have the same dimensions.

How many forces are acting on the ball? What must be true about them if the ball is in equilibrium (i.e. not accelerating)? This is basic physics that is covered well before electromagnetism.

 

[bobh.stein]

So you do the derivative in terms of z and then solve that equal to zero for part a and then put it back into the original equation.

For part B I'm guessing the electric force which is given by F=EQ has to be equal to the weight which is MG. Then i would rearrange so that it is M=EQ/G and then sub the first equation in for the value of E and then differentiate and find the maximum.

 

[cepheid]

So you do the derivative in terms of z and then solve that equal to zero for part a and then put it back into the original equation.

For part B I'm guessing the electric force which is given by F=EQ has to be equal to the weight which is MG. Then i would rearrange so that it is M=EQ/G and then sub the first equation in for the value of E and then differentiate and find the maximum.

You got it!

For part b, I don't think you need to do any differentiating. All you have to do is find the value of mg that is equal to the electric force *at the point of maximum electric field strength.* This automatically tells you the largest mass that can be supported. Lighter masses will be pushed up until the electric field weakens and balance is achieved. Heavier masses will sink.

 

[bobh.stein]

Thank you for your help, so to find the maximum electric force do i take my value for E(z) from part a because that is the maximum and then i have to come up with a maximum value for Q. I'm not sure exactly sure how to go about doing this.

 

[cepheid]

Thank you for your help, so to find the maximum electric force do i take my value for E(z) from part a because that is the maximum and then i have to come up with a maximum value for Q. I'm not sure exactly sure how to go about doing this.

No, Q is a constant. It's just the value for the charge on the ball. *Regardless* of what Q is, the force on the ball will be a maximum when it is at the z-position where the electric field due to the ring is a maximum.

 

[bobh.stein]

Is the maximum z position just the amswer that i obtained in part a?