Calculate the required current flow in the suspended wire

[Taiyen]

Homework Statement

In the figure, the top wire is 1.5 mm diameter copper wire and is suspended in air due to the two magnetic forces from the bottom two wires. The current flow through the two bottom wires is 70 A in each.

Calculate the required current flow in the suspended wire.

Homework Equations

F=(I₁I₂μ_o*length)/2πr
F=(r²πρ)/length

The Attempt at a Solution

I canceled length and now have combined:

(r²πρ)=(I₁I₂μ_o)/2πr

I₁ = 70A
ρ (density of copper) = 8.92g/cm²
μ_o = 4π*10^-3
r (on the left side) = .15cm/2
r (on the right side) = 3.8cm

I just plug these in and find I₂...?
I have tried to, but I keep getting the wrong answer.. maybe my r's are mixed up? I tried switching them, but still wrong.. I think I am missing something, maybe my wrote down the equations incorrectly.. please help! I don't need the final answer, I'm more interested in learning how to do this type of problem.

Thank you in advance!

 

[TSny]

Welcome to PF!

F=(I₁I₂μ_o*length)/2πr
F=(r²πρ)/length

Did you mean for the length in the second equation to be in the numerator? Does the acceleration due to gravity play any role here?

Think about the direction of the magnetic force on the the top wire due to each of the bottom wires.

Since you didn't show your calculation, we can't tell if you did the unit conversions correctly. (Your units for the density of copper are incorrect.)

What units are you using for expressing μ_o?

 

[Taiyen]

Oh, yes! The length on the left side is supposed to be in the numerator to cancel. That is my mistake!

I thought that I could calculate mg (in the equation F=mg) by using r²πρ*length, but I really am not sure that this is correct..
Would the direction be downward?

μ₀'s units are Wb/m².

And again, my mistake, the density of copper's units are in kg/m³ not g/cm².

I really appreciate your help!

 

[Taiyen]

Since you didn't show your calculation, we can't tell if you did the unit conversions correctly.

Here's what I have after I converted everything to meters:

(0.0015m/2)²*π*(8.92kg/m³) = [(70A) * I₂ * (4π*10^-3 Wb/m²)]/(2π*0.038m)
and I found I₂ to be 4.27*10^-6, so I am doing something very wrong...

 

[TSny]

Your value for the density in kg/m³ is incorrect. Be careful with converting cm³ to m³. Also, your value for μ_o in the SI system of units is incorrect. Check your textbook or notes.

There are two magnetic forces on the upper wire; one force from each of the lower wires. These forces are vector quantities and so you need to consider the directions of these forces.

 

[TSny]

I thought that I could calculate mg (in the equation F=mg) by using r²πρ*length, but I really am not sure that this is correct..
Would the direction be downward?

What happened to the "g"? Yes, the direction is downward.

μ₀'s units are Wb/m²

These are not correct units for μ₀.

 

[Taiyen]

What happened to the "g"? Yes, the direction is downward.These are not correct units for μ₀.

OHH! Okay okay! I think I get it!
Sorry for the many mistakes in values and units, I looked back in the textbook and corrected everything.
μ_o = 4π*10^-7 Tm/A
density value is 8960 kg/m³

And then with 2*Fcos(30) = ρπr²*length*g, I can cancel lengths, the "2", and put it back into the other equation to get:

ρπr²g = [I₁I₂μ_ocos(30)]/(πr)

I got my correct answer with this, thank you so very much for your help! c: I really appreciate it!

 

[TSny]

Great. Good work!