Solveeit Logo

Question

Question: How many meters of a thin wire is required to design a solenoid of length \(1\,m\) and \(L = 1\,mH\)...

How many meters of a thin wire is required to design a solenoid of length 1m1\,m and L=1mHL = 1\,mH? Assume the cross-sectional diameter is very small.
A. 10m10\,m
B. 40m40\,m
C. 70m70\,m
D. 100m100\,m
E. 140m140\,m

Explanation

Solution

We know that inductance of solenoid is given as,
L=μ0N2AlL = \dfrac{{{\mu _0}{N^2}A}}{l}
Where NN is the number of turns in the solenoid. AA is the area of each turn ll is the length of the solenoid and μ0{\mu _0} is a constant whose value is 4π×1074\pi \times {10^{ - 7}} .
We can express the number of turns in terms of length of the wire as
N=x2πrN = \dfrac{x}{{2\pi r}}
Where xx denotes the length of wire and the circumference of each turn is 2πr2\pi r
Substituting this in the first equation we can find the length of wire.

Complete step by step answer:
The inductance of a solenoid is given by the formula
L=μ0N2AlL = \dfrac{{{\mu _0}{N^2}A}}{l} …………………..(1)
Where NN is the number of turns in the solenoid. AA is the area of each turn. And ll is the length of the solenoid and μ0{\mu _0} is a constant whose value is 4π×1074\pi \times {10^{ - 7}}.
The length of the wire, xx of the solenoid is circumference of each turn multiplied by the number of turns. That is,
x=2πr×Nx = 2\pi r \times N
Since, circumference of a circle is 2πr2\pi r
Therefore, the value of NN is given by
N=x2πrN = \dfrac{x}{{2\pi r}}
Area of the circle is
A=πr2A = \pi {r^2}
Substituting all these values in equation 1 we get
We get,
L=4π×107(x2πr)2πr2lL = \dfrac{{4\pi \times {{10}^{ - 7}}{{\left( {\dfrac{x}{{2\pi r}}} \right)}^2}\pi {r^2}}}{l}
L=4π×107×x2×πr24π2r2l\Rightarrow L = \dfrac{{4\pi \times {{10}^{ - 7}} \times {x^2} \times \pi {r^2}}}{{4{\pi ^2}{r^2}l}}
x=l×L107\Rightarrow x = \sqrt {\dfrac{{l \times L}}{{10 - 7}}}
Given value of l=1ml = 1\,m and L=1mHL = 1mH
On Substituting we get

x=1×1×103107 x=100m \Rightarrow x = \sqrt {\dfrac{{1\, \times 1 \times {{10}^{ - 3}}\,}}{{{{10}^{ - 7}}}}} \\\ \Rightarrow x = 100\,m \\\

Therefore, the correct option is (D).

Note:
Here, we took the turns of the solenoid to be circular because it is given that the cross-sectional diameter of the solenoid is very small. That is why in the equation for the number of turns we divided the total length of the wire by circumference of the circle, 2πr2\pi r. But for large cross-sectional diameter, we cannot assume each turn as a circle.