Question
A 300 turn coil has a resistance of 6ohms and an inductance of 0.5 H
Determine the new resistance and the new inductance if one third of the turns are removed . Assume all the turns have the same circumference
Determine the new resistance and the new inductance if one third of the turns are removed . Assume all the turns have the same circumference
Answers
Answered by
GPT-4o mini
To determine the new resistance and inductance of the coil after removing one third of the turns, we can follow these steps:
### Step 1: Calculate the new number of turns
Original number of turns, \( N = 300 \).
If one third of the turns are removed, the number of turns left is:
\[
N' = N - \frac{1}{3}N = \frac{2}{3}N = \frac{2}{3} \times 300 = 200
\]
### Step 2: Calculate the new resistance
The resistance of a coil is directly proportional to the number of turns, assuming the wire is of the same material and cross-sectional area. The formula for resistance is:
\[
R' = R \times \frac{N'}{N}
\]
Where:
- \( R \) is the original resistance (6 ohms),
- \( R' \) is the new resistance,
- \( N' \) is the new number of turns (200),
- \( N \) is the original number of turns (300).
Substituting the values:
\[
R' = 6 \, \Omega \times \frac{200}{300} = 6 \, \Omega \times \frac{2}{3} = 4 \, \Omega
\]
### Step 3: Calculate the new inductance
The inductance of a coil is proportional to the square of the number of turns. The formula for inductance is:
\[
L' = L \times \left(\frac{N'}{N}\right)^2
\]
Where:
- \( L \) is the original inductance (0.5 H),
- \( L' \) is the new inductance.
Substituting the values:
\[
L' = 0.5 \, \text{H} \times \left(\frac{200}{300}\right)^2 = 0.5 \, \text{H} \times \left(\frac{2}{3}\right)^2 = 0.5 \, \text{H} \times \frac{4}{9}
\]
Calculating the above:
\[
L' = 0.5 \, \text{H} \times \frac{4}{9} \approx 0.222 \, \text{H}
\]
### Summary of Results
- **New Resistance \( R' \)**: \( 4 \, \Omega \)
- **New Inductance \( L' \)**: \( \approx 0.222 \, \text{H} \)
Thus, after removing one third of the turns, the new resistance is 4 ohms and the new inductance is approximately 0.222 H.
### Step 1: Calculate the new number of turns
Original number of turns, \( N = 300 \).
If one third of the turns are removed, the number of turns left is:
\[
N' = N - \frac{1}{3}N = \frac{2}{3}N = \frac{2}{3} \times 300 = 200
\]
### Step 2: Calculate the new resistance
The resistance of a coil is directly proportional to the number of turns, assuming the wire is of the same material and cross-sectional area. The formula for resistance is:
\[
R' = R \times \frac{N'}{N}
\]
Where:
- \( R \) is the original resistance (6 ohms),
- \( R' \) is the new resistance,
- \( N' \) is the new number of turns (200),
- \( N \) is the original number of turns (300).
Substituting the values:
\[
R' = 6 \, \Omega \times \frac{200}{300} = 6 \, \Omega \times \frac{2}{3} = 4 \, \Omega
\]
### Step 3: Calculate the new inductance
The inductance of a coil is proportional to the square of the number of turns. The formula for inductance is:
\[
L' = L \times \left(\frac{N'}{N}\right)^2
\]
Where:
- \( L \) is the original inductance (0.5 H),
- \( L' \) is the new inductance.
Substituting the values:
\[
L' = 0.5 \, \text{H} \times \left(\frac{200}{300}\right)^2 = 0.5 \, \text{H} \times \left(\frac{2}{3}\right)^2 = 0.5 \, \text{H} \times \frac{4}{9}
\]
Calculating the above:
\[
L' = 0.5 \, \text{H} \times \frac{4}{9} \approx 0.222 \, \text{H}
\]
### Summary of Results
- **New Resistance \( R' \)**: \( 4 \, \Omega \)
- **New Inductance \( L' \)**: \( \approx 0.222 \, \text{H} \)
Thus, after removing one third of the turns, the new resistance is 4 ohms and the new inductance is approximately 0.222 H.