A 200 turn coil has an inductance of 12 mH. If the number of turns is increased to 400 turns, all other quantities (area, length etc.) remaining the same, the inductance will be?
Understand the Problem
The question is asking about the relationship between the number of turns in a coil and its inductance. It provides an initial condition and asks to calculate the new inductance when the number of turns is increased, while keeping other factors constant.
Answer
The new inductance is \( 48 \, \text{mH} \).
Answer for screen readers
The new inductance is ( 48 , \text{mH} ).
Steps to Solve
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Understand the formula for inductance The inductance ( L ) of a coil is related to the square of the number of turns ( N ) in the coil as follows: $$ L \propto N^2 $$ This means that if the number of turns is doubled, the inductance increases by a factor of four.
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Calculate the new number of turns The initial number of turns is ( N_1 = 200 ) turns and the new number of turns is ( N_2 = 400 ) turns.
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Find the ratio of the turns To find the relationship: $$ \text{Ratio of turns} = \frac{N_2}{N_1} = \frac{400}{200} = 2 $$
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Find the new inductance Since the inductance scales with the square of the number of turns: $$ L_2 = L_1 \times \left( \frac{N_2}{N_1} \right)^2 $$ Substituting the known values: $$ L_2 = 12 , \text{mH} \times 2^2 = 12 , \text{mH} \times 4 = 48 , \text{mH} $$
The new inductance is ( 48 , \text{mH} ).
More Information
When you double the number of turns in a coil, the inductance quadruples. This relationship is essential in electromagnetic theory and applications.
Tips
- Ignoring the square relationship: It is common to mistakenly think the inductance simply doubles with an increase in turns. Remember that it increases with the square of the turns.
- Incorrect calculations: Watch for miscalculations when applying the ratio of turns to find the new inductance.
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