Question

You are required to build a solenoid with resistance, R = 8 ohms. In order to do this, you will use a copper cable with resistivity p= 1.7*10-8 1.m. The cross sectional diameter of the cable is of 0.5 mm and it will be wrapped around a cylindrical conductor to build the solenoid with radius, R = 6 cm. The magnetic field at the center of the solenoid is B = 5.0*10-4 T when a current I = 5 Amperes is passing through. Your task is to compute the number of turns around the solenoid and the length this solenoid must have.(12 points)

Answer 1

Answer:

Given, resistance of the cable R = 8 $\Omega$ ,

resistivity of the copper $\rho$ = 1.7 x 10^-8$\Omega$.m,

diameter of the cable d = 0.5 mm, so its radius is r = d/2 = 0.25 x 10^-3 m,

radius of the cylindrical conductor R = 6 cm = 0.06 m,

B = 5.0 x 10^-4 T and

current I = 5 A

Resistance R = $\rho$ L/A = $\rho$ L/$\pi$r²

Thus, length of the cable is L = R$\pi$r²/$\rho$ = (8 $\Omega$ )$\pi$(0.25 x 10^-3 m)² / (1.7 x 10^-8$\Omega$.m) = 92.35 m

and N = L/2$\pi$R

where 2$\pi$R is the curcumference of the cylindrical conductor

Then, N = (92.35 m)/2$\pi$(0.06 m) = 245.61$\approx$245 turns

The magnetic field due to the solenoid is B = μ₀nI

Thus, n = B/ μ₀I = (5.0 x 10^-4 T) / (4$\pi$ x 10^-7 T.m/A) (5 A) = 79.61 turns/m $\approx$ 79 turns/m

Therefore, n = N/L'

where L' is the length of the solenoid

L' = N/n = (245 turns )(79 turns/m) = 3.10 m

Hence, number of tuns N= 245 and the length of the solenoid is L' = 3.10 m