To solve the problem, we will perform two calculations: the resistance of the winding and the average value of the induced emf when the current is reversed.
1) Calculate the resistance of the winding
The resistance \( R \) of the coil can be calculated using the formula:
\[ R = \rho \frac{L}{A} \]
where:
- \( \rho \) = specific resistance of the copper wire = \( 0.017 \times 10^{-6} , \Omega \cdot m \)
- \( L \) = total length of the wire (in meters)
- \( A \) = cross-sectional area of the wire (in \( m^2 \))
First, we need to find the total length of the wire. The mean length per turn is given as 25 cm (which is \( 0.25 , m \)), and the number of turns is 1200.
Thus, the total length \( L \) is:
\[ L = \text{number of turns} \times \text{mean length per turn} = 1200 \times 0.25 , m = 300 , m \]
Next, convert the cross-sectional area from \( mm^2 \) to \( m^2 \):
\[ A = 2.5 , mm^2 = 2.5 \times 10^{-6} , m^2 \]
Now we can substitute into the resistance formula:
\[ R = 0.017 \times 10^{-6} \cdot \frac{300}{2.5 \times 10^{-6}} \]
Calculating:
\[ R = 0.017 \times 10^{-6} \cdot \frac{300}{2.5 \times 10^{-6}} = 0.017 \cdot \frac{300}{2.5} = 0.017 \cdot 120 = 2.04 \times 10^{-2} , \Omega \]
Thus, the resistance of the winding is:
\[ R \approx 0.0204 , \Omega \]
2) Calculate the average value of the emf induced in the coil
The emf induced in the coil can be calculated using Faraday’s law of electromagnetic induction, given by:
\[ \text{emf} = -L \frac{\Delta I}{\Delta t} \]
where:
- \( L \) = inductance of the coil = \( 48 \times 10^{-6} , H \)
- \( \Delta I \) = change in current = \( I_{final} - I_{initial} \) (Since the current is reversed, we take the change from \( +6.5 , A \) to \( -6.5 , A \))
- \( \Delta t \) = time duration of the change = \( 1.5 , ms = 1.5 \times 10^{-3} , s \)
Calculating \( \Delta I \):
\[ \Delta I = -6.5 - 6.5 = -13 , A \]
Now substitute the values into the emf formula:
\[ \text{emf} = -48 \times 10^{-6} \cdot \frac{-13}{1.5 \times 10^{-3}} \]
Calculating:
\[ \text{emf} = 48 \times 10^{-6} \cdot \frac{13}{1.5 \times 10^{-3}} = 48 \times 10^{-6} \cdot \frac{13 \times 10^3}{1.5} = 48 \times 10^{-6} \cdot 8666.67 \]
\[ \text{emf} = 0.416 , V \]
Thus, the average value of the emf induced in the coil is approximately:
\[ \text{emf} \approx 0.416 , V \]
Summary of Results
-
The resistance of the winding: \( \approx 0.0204 , \Omega \)
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The average value of the emf induced: \( \approx 0.416 , V \)