You are designing a solenoid to produce a 2.0-kG magnetic field. You wish to wrap your insulated wire uniformly around a cardboard tube that is 8.0 cm in diameter and 58 cm in length, and you have a power supply that will allow you to pass a current of 2.5 A through the solenoid. Determine the total length of wire you will need in order to build the solenoid you have designed.

Since the magnetic field due to a solenoid is given by B = μ₀Ni/L where μ₀ = permeability of free space = 4π × 10⁻⁷ H/m, N = number of turns of solenoid, L = length of cardboard tube = 58 cm = 0.58 m, , i = current in wire = 2.5 A and l = length of wire.

So, N = BL/μ₀i

Since B = 2.0 kG = 2.0 × 10³ G = 2.0 × 10³ × 10⁻⁴ T = 2.0 × 10⁻¹ T = 0.2 T

So, substituting the variables into the equation, we have

N = BL/μ₀i

N = 0.2 T × 0.58 m/(4π × 10⁻⁷ H/m × 2.5 A)

N = 1.16 Tm/(31.416 × 10⁻⁷ HA/m)

N = 0.0369 × 10⁷ turns

N = 3.69 × 10⁵ turns

length of wire l = NC where N = number of turns and C = circumference of tube = πD where D = diameter of tube = 8.0 cm = 0.08 m

So, l = NC

= NπD

= πND

= π × 3.69 × 10⁵ turns × 0.08 m

= 0.9274 × 10⁵ m = 9.274 × 10⁴ m

= 92.74 × 10³ m

= 92.74 km

≅ 92.7 km

shubhamchouhanvrVT shubhamchouhanvrVT · Answer 2 · 2022-04-09T09:55:30+02:00

The total legnth of the wire to built a solenoid will be 92.7 km

What is solenoid?

A solenoid is a electromagnet in which the wires are wounded on the outer surface of the soft iron and then the current is passed so due to flow of the current the magnetic field generated around the wire.

Since the magnetic field due to a solenoid is given by

B = μ₀Ni/L

where

μ₀ = permeability of free space = 4π × 10⁻⁷ H/m,

N = number of turns of solenoid,

L = length of cardboard tube = 58 cm = 0.58 m, ,

i = current in wire = 2.5 A and

l = length of wire.

Since B = 2.0 kG = 2.0 × 10³ G = 2.0 × 10³ × 10⁻⁴ T = 2.0 × 10⁻¹ T = 0.2 T

So, substituting the variables into the equation, we have

[tex]N=\dfrac{BL}{\mu_oI}[/tex]

[tex]N=\dfrac{0.2\times 0.58}{4\pi\times 10^{-7} \times 2.5}[/tex]

[tex]N=\dfrac{1.16}{31.416\times 10^{-7}}[/tex]