To solve your problem, we'll tackle it step by step.
Step 1: Calculate the Resistance
To calculate the resistance \( R \) of the coil, we can use the formula:
\[ R = \rho \frac{L}{A} \]
where:
- \( R \) is the resistance,
- \( \rho \) is the resistivity of the wire,
- \( L \) is the length of the wire,
- \( A \) is the cross-sectional area of the wire.
Given values:
- \( \rho = 0.027 , \mu\Omega , m = 0.027 \times 10^{-6} , \Omega, m \)
- Diameter of the wire = 1.2 mm = 0.0012 m
- Mean length \( L = 120 , mm = 0.12 , m \)
- \( N = 440 , turns \)
Calculate \( A \):
The cross-sectional area \( A \) of the wire can be calculated using the formula for the area of a circle:
\[ A = \pi \left(\frac{d}{2}\right)^2 = \pi \left(\frac{0.0012}{2}\right)^2 \]
Calculating it:
\[ A = \pi \left(0.0006\right)^2 \approx \pi \times 0.00000036 \approx 1.131 \times 10^{-6} , m^2 \]
Calculate \( R \):
Now substituting \( \rho \), \( L \), and \( A \) into the resistance formula:
\[ R = 0.027 \times 10^{-6} \frac{0.12}{1.131 \times 10^{-6}} \approx 0.027 \times 10^{-6} \times 106,272.5 \]
Calculating it:
\[ R \approx 0.027 \times 0.000028707 \approx 0.000000773 , \Omega = 0.773 , \mu\Omega \]
Step 2: Calculate the Average EMF Induced
The average EMF \( \mathcal{E} \) induced in the coil can be calculated using Faraday's law of electromagnetic induction:
\[ \mathcal{E} = -L \frac{\Delta I}{\Delta t} \]
Where:
- \( L \) is the inductance (in henries),
- \( \Delta I \) is the change in current,
- \( \Delta t \) is the change in time.
Given values:
- \( L = 400 , \mu H = 400 \times 10^{-6} , H \)
- Change in current \( \Delta I = 4A - (-4A) = 4A + 4A = 8A \)
- Change in time \( \Delta t = 0.001, s \)
Calculate \( \mathcal{E} \):
Substituting the values:
\[ \mathcal{E} = -400 \times 10^{-6} \frac{8}{0.001} \]
Calculating it:
\[ \mathcal{E} = -400 \times 10^{-6} \times 8000 = -3.2 , V \]
Thus, the average value of the EMF induced in the coil is approximately 3.2 V (the negative sign indicates the direction of induced EMF opposing the change in current).
Final Results:
- Resistance of the coil: \( R \approx 0.773 , \mu\Omega \)
- Average EMF induced: \( |\mathcal{E}| \approx 3.2 , V \)