Question 217: Electromagnetic Induction

Question

A coil is wound on a core of rectangular crosssection. If all the linear dimensions of core are increased by a factor 3 and number of turns per unit length of coil remains same, the selfinductance increases by a factor

Accepted Answer

The self-inductance of a coil (approximated as a solenoid wound on a core) is given by the formula:
$L = \mu n^2 A l$
Where:
$L$ is the self-inductance.
$\mu$ is the permeability of the core material.
$n$ is the number of turns per unit length.
$A$ is the cross-sectional area of the core.
$l$ is the length of the coil (which is the length of the core along the axis of the coil).
Let's analyze how each parameter changes based on the problem statement:
Linear dimensions of the core
: The problem states that "all the linear dimensions of core are increased by a factor 3".
A core with a rectangular cross-section has three main linear dimensions: width ($w$), height ($h$), and length ($l$).
So, the new dimensions will be:
New width $w' = 3w$
New height $h' = 3h$
New length $l' = 3l$
Cross-sectional area ($A$)
:
The original cross-sectional area was $A = w 	imes h$.
The new cross-sectional area $A'$ will be $A' = w' 	imes h' = (3w) 	imes (3h) = 9wh = 9A$.
So, $A'$ increases by a factor of 9.
Length of the coil ($l$)
:
As part of the linear dimensions of the core, the length of the coil (the length over which the turns are wound) also increases by a factor of 3.
So, $l' = 3l$.
Number of turns per unit length ($n$)
:
The problem explicitly states that "number of turns per unit length of coil remains same".
So, $n' = n$.
Permeability ($\mu$)
:
The core material is not changed, so its permeability $\mu$ remains constant.
Now, let's calculate the new self-inductance ($L'$):
The original self-inductance was $L = \mu n^2 A l$.
The new self-inductance is $L' = \mu (n')^2 A' l'$.
Substitute the changed parameters:
$L' = \mu (n)^2 (9A) (3l)$
$L' = \mu n^2 A l 	imes (9 	imes 3)$
$L' = \mu n^2 A l 	imes 27$
Since $L = \mu n^2 A l$, we can write:
$L' = 27L$
Therefore, the self-inductance increases by a factor of 27.
The final answer is $\boxed{27}$.

Answer

$\frac{1}{6}$

Answer

1

Answer

3

Answer

27

Electromagnetic Induction

A coil is wound on a core of rectangular crosssection. If all the linear dimensions of core are increased by a factor 3 and number of turns per unit length of coil remains same, the selfinductance increases by a factor

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