A coil is wound on a core of rectangular cross section. 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 self inductance increases by a factor
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The self-inductance of a coil depends on the cross-sectional area and the length of the coil. If both dimensions are scaled by a factor, the inductance changes by the square of the scaling factor for area and the scaling factor for length.
Step 1: Understanding the Question:
Self-inductance \( L \) depends on the geometry of the coil. We need to see how scaling all dimensions affects \( L \) when turns per unit length is constant. Step 2: Key Formula or Approach:
Self-inductance of a solenoid (or similar coil): \( L = \mu_0 n^2 A l \).
Where:
\( n = \) turns per unit length
\( A = \) cross-sectional area
\( l = \) length of the coil Step 3: Detailed Explanation:
Given turns per unit length \( n \) is constant.
Linear dimensions are increased by factor \( k = 3 \).
New length \( l' = 3l \).
New area \( A' \propto (\text{linear dimension})^2 \). So \( A' = 3^2 A = 9A \).
New self-inductance:
\[ L' = \mu_0 n^2 (9A) (3l) \]
\[ L' = 27 (\mu_0 n^2 A l) = 27L \] Step 4: Final Answer:
The self-inductance increases by a factor of 27.