Answer:
![P=214.7187\,W](https://tex.z-dn.net/?f=P%3D214.7187%5C%2CW)
Explanation:
Given that:
Diameter of the solenoid, ![D=10\,cm=0.1\,m](https://tex.z-dn.net/?f=D%3D10%5C%2Ccm%3D0.1%5C%2Cm)
length of the solenoid, ![L=90\,cm=0.9\,m](https://tex.z-dn.net/?f=L%3D90%5C%2Ccm%3D0.9%5C%2Cm)
diameter of the wire, ![d=0.1\,cm=10^{-3}\,m](https://tex.z-dn.net/?f=d%3D0.1%5C%2Ccm%3D10%5E%7B-3%7D%5C%2Cm)
magnetic field at the center of the solenoid, ![B=7.4\times 10^{-3}\,T](https://tex.z-dn.net/?f=B%3D7.4%5Ctimes%2010%5E%7B-3%7D%5C%2CT)
<u>Now we need the no. of turns incorporated in the length of 90 cm:</u>
![N=\frac{Length\,\,of\,\,solenoid}{diameter\,\,of\,\, wire}](https://tex.z-dn.net/?f=N%3D%5Cfrac%7BLength%5C%2C%5C%2Cof%5C%2C%5C%2Csolenoid%7D%7Bdiameter%5C%2C%5C%2Cof%5C%2C%5C%2C%20wire%7D)
![N=\frac{L}{d}](https://tex.z-dn.net/?f=N%3D%5Cfrac%7BL%7D%7Bd%7D)
![N=\frac{0.9}{10^{-3}}](https://tex.z-dn.net/?f=N%3D%5Cfrac%7B0.9%7D%7B10%5E%7B-3%7D%7D)
![N=900\,\,turns](https://tex.z-dn.net/?f=N%3D900%5C%2C%5C%2Cturns)
For solenoids we have:
...............................(1)
where:
permeability of the medium
n = no. of turns per unit length
I = current in the coil
So,
![n=\frac{900}{0.9}](https://tex.z-dn.net/?f=n%3D%5Cfrac%7B900%7D%7B0.9%7D)
![n=1000\,turns\,.\,m^{-1}](https://tex.z-dn.net/?f=n%3D1000%5C%2Cturns%5C%2C.%5C%2Cm%5E%7B-1%7D)
Now putting the respective values in the eq. (1)
![7.4\times 10^{-3}=4\pi\times10^{-7}\times 1000\times I](https://tex.z-dn.net/?f=7.4%5Ctimes%2010%5E%7B-3%7D%3D4%5Cpi%5Ctimes10%5E%7B-7%7D%5Ctimes%201000%5Ctimes%20I)
![I=5.8887\,A](https://tex.z-dn.net/?f=I%3D5.8887%5C%2CA)
- For copper we have resistivity:
We know that resistance is given by:
.....................................(2)
where:
l = length of the conducting wire
a = cross sectional area of the conducting wire
<u>Now we need the length (l) of the wire:</u>
Circumference of the solenoid,
![C=\pi.D](https://tex.z-dn.net/?f=C%3D%5Cpi.D)
![C=0.1\pi\,m](https://tex.z-dn.net/?f=C%3D0.1%5Cpi%5C%2Cm)
![\therefore l=C\times N](https://tex.z-dn.net/?f=%5Ctherefore%20l%3DC%5Ctimes%20N)
![l=90\pi\,m](https://tex.z-dn.net/?f=l%3D90%5Cpi%5C%2Cm)
&
<u>Cross-sectional area of wire:</u>
![a=\pi.\frac{d^2}{4}](https://tex.z-dn.net/?f=a%3D%5Cpi.%5Cfrac%7Bd%5E2%7D%7B4%7D)
![a=\pi. \frac{(10^{-3})^2}{4}\,m^2](https://tex.z-dn.net/?f=a%3D%5Cpi.%20%5Cfrac%7B%2810%5E%7B-3%7D%29%5E2%7D%7B4%7D%5C%2Cm%5E2)
<u>Resistance from eq. (2):</u>
![R=1.72\times 10^{-8}\times \frac{90\pi}{\pi. \frac{(10^{-3})^2}{4}}](https://tex.z-dn.net/?f=R%3D1.72%5Ctimes%2010%5E%7B-8%7D%5Ctimes%20%5Cfrac%7B90%5Cpi%7D%7B%5Cpi.%20%5Cfrac%7B%2810%5E%7B-3%7D%29%5E2%7D%7B4%7D%7D)
![R=6.192 \,\Omega](https://tex.z-dn.net/?f=R%3D6.192%20%5C%2C%5COmega)
![P=I^2.R](https://tex.z-dn.net/?f=P%3DI%5E2.R)
![P=5.8887^2 \times 6.192](https://tex.z-dn.net/?f=P%3D5.8887%5E2%20%5Ctimes%206.192)
![P=214.7187\,W](https://tex.z-dn.net/?f=P%3D214.7187%5C%2CW)