Current on a spring to withstand a weight.

[pitbull]

Homework Statement

You have a spring of length l, radius R, with N loops and n loops per unit length. If you consider it a solenoid, what current do you need to apply to withstand a mass m hanging from it, without stretching or shrinking the spring.

Homework Equations

Magnetic field inside a solenoid: B=μ₀nIz
Magnetic energy: W=∫∫∫B²/(2μ₀)dV

The Attempt at a Solution

I calculated the magnetic energy of a solenoid and I got:
W=μ₀N²I²πR²/(2l)

The force applied by the mass is:
F=mg

So, the magnetic force that I need is:
F=∇W(length constant)=μ₀N²IπR²/(l)

Both forces must be equal. I solve for I and I get:
I=mgl/(μ₀N²πR²)

Do you thing it is right?

 

[mfb]

The approach looks good, but I get confused by the notation with I, l and l.
The derivative has to be with respect to length.

 

[pitbull]

The approach looks good, but I get confused by the notation with I, l and l.
The derivative has to be with respect to length.

So should I derivate with respect to length and then equal that derivative to mg?

 

[mfb]

Sure. You are interested in the total energy after a length change to see if this length change happens.

 

[pitbull]

Sure. You are interested in the total energy after a length change to see if this length change happens.

And after equaling both quantities, I solve for I (current), right?

 

[mfb]

Sure.