If you have a few hundred dollars on hand and a keen interest in magnets, you ca

Question

If you have a few hundred dollars on hand and a keen interest in magnets, you can buy a
neodymium cylinder magnet (5 cm diameter, 5 cm height) with a magnetic field strength of
0.6 T or more at its surface. The magnet is small enough to hold in your hand, but you'd have
to be very careful to keep your hand far from any other metal object; if one of these gets stuck
to your refrigerator, you'd have to exert a force of about 400 lbs to pull it off!

Question: if you made a solenoid the same size as this neodymium magnet, could you
achieve the same field strength at a reasonable operating power?
The solenoid would be wound with very, very thin wire (magnet wire) to allow the maximum
number of turns per unit length, since that and the current determine the field strength. One of
the thinnest magnet wire diameters is 0.10 mm, so assume you have a sufficient supply of
copper wire with that diameter. Imagine winding one layer of this wire around a wooden
cylinder that's exactly the same size as the neodymium magnet, making sure there are no
gaps between neighboring turns. When the ends of the wire are connected to a power supply,
this is your solenoid equivalent of the neodymium magnet.
To determine the power needed to operate the solenoid, you'll need to figure out the number
of turns per meter, the required current, the total length of the wire, and then the resistance of
the wire used to create the solenoid.
It may be confusing at first to figure out how many turns per unit length there will be, but think
about it this way: if the diameter of the wire were 1 meter, a coil constructed of the wire would
have just 1 turn per meter. If the wire diameter were 10 cm = 0.1 m = 1/10 of a meter, a coil
made of it would have 10 turns per meter, since ten wire diameters would make up a meter.
And if the wire diameter were 1 cm = 0.01 m = 1/100 of a meter, a coil made of it would have
100 turns per meter. (And note that “turns per unit length” means “turns per unit length of the
solenoid (cylinder)” rather than “turns per unit length of wire”.)
For this problem, ignore temperature effects: assume the wire remains at room temperature,
even if you get a large power output.

Explanation / Answer

since the wire diameter is 0.1mm, or 1/10000 of a meter, the solenoid will have 10000 windings per meter (this is "n").

And... the total length of the wire would be:   circumference * number of turns

    = 2 pi R * n * length = 2 pi * 2.5cm * 10000 turns per m * 0.05 m = 7854 cm = 78.54 meters

the current required, using equation for a solenoid, is

   B = u n I or      0.6 = 4pi x 10^-7 * 10000 * I

       I = 47.74 amps

The resistance of the wire is   resistivity of copper * length / area =

   = resistivity * length / pi r^2 = 1.72x10^-8 * 78.54 / pi * 0.00005^2 = 172 ohms

Finally, the power required to supply this solenoid is   current squared * resistance or

    47.74^2 * 172 =    392000 Watts

Probably not worth building this solenoid. Just buy the magnet.

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