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.