Question

If all the linear dimensions of a cylindrical coil are doubled, the inductance of the coil will be (assuming complete winding over the core and the diameter of wire remains unchanged)
A. doubled
B. Four fold
C. Eight times
D. Remains unchanged

Answer 1

Hint: In question, length and inner core area would get doubled, whole numbers of turns would remain the same. We have a direct relationship between dimensions of coil and inductance, thus by substituting twice the value of length and cross-sectional area, effect on inductance can be calculated.

Complete step by step answer:
Let us consider an inductor in form of a coil wound around a cylinder of length ‘l’ and area of cross-section ‘A’ as shown in figure: Given in question that, linear dimensions of the cylindrical coil are doubled that means length and area of cross section of inner core is doubled whereas complete winding over the core is unchanged that means number of turns are the same as before. Usually, linear dimensions are physical quantities which define the measurement of any object. For example, in case of a cuboidal box, the linear dimensions would be length, breadth and height. Coming to the question,
\[
l \to {\text{2}}l \\
\Rightarrow{\text{A}} \to {\text{2A}} \\ \]
We are asked to find the inductance of the coil, for that we need to have a direct relationship between dimensions of coil and inductance and we have already derived it in an article of “self-Inductance of a long solenoid”. It may be given as:
\[L = \dfrac{{{\mu _0}{N^2}A}}{l}\] ………………………………………Eq. 1
Where, L= Inductance of coil/solenoid (SI unit for inductance is Henry)
\[{\mu _0}\]= magnetic permeability of free space (value of \[{\mu _0}\] =\[4\pi \times {10^{ - 7}}\])
N=total number of turns
A= area of cross-section of coil
l=length of coil
Putting \[l = 2l\] and \[A = 2A\]in Eq.1
We get,
\[ \Rightarrow L' = \dfrac{{{\mu _0}{N^2}2A}}{{2l}}\]
\[ \Rightarrow L' = \dfrac{{{\mu _0}{N^2}A}}{l}\] ……………………………Eq.2
Here, \[L'\]= new inductance of coil.
From eq.1 and eq.2, we get that
\[ \therefore L = L'\]
Because, we get the resultant equation same as equation 1. Therefore, the inductance of the coil will remain unchanged.

Hence, option D is correct.

Note:From the above relationship, it can be concluded that inductance (L) of a coil/solenoid is inversely proportional to the length of wire and directly proportional to area of cross-section of inner core if number of turns is kept constant. Thus, if only the length of coil was increased, then inductance would have been decreased.