Question
A coil having :
N= 440 turns
diameter =1,2mm
Mean length= 120mm
Inductance= 400micro Henry's
Calculate the resistance is the resistivity of the wire is 0,027micro-ohm meters
Calculate the average value of the EMF induced in a coil when current of 4A is reversed in 0,001 seconds
N= 440 turns
diameter =1,2mm
Mean length= 120mm
Inductance= 400micro Henry's
Calculate the resistance is the resistivity of the wire is 0,027micro-ohm meters
Calculate the average value of the EMF induced in a coil when current of 4A is reversed in 0,001 seconds
Answers
GPT-4o mini
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**: \( |\mathcal{E}| \approx 3.2 \, V \)
### 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**: \( |\mathcal{E}| \approx 3.2 \, V \)