A magnetic field given by B(t) `=0.2t-0.05t^(2)` tesla is directed perpendicular to the plane of a circular coil containing 25 turns of radius 1.8 cm and whose total resistance is `1.5Omega`. The power dissipation at 3 s is approximately

Options

A. `1.37muW`
B. `7muW`
C. zero
D. `4muW`

Explanation

To solve the problem step by step, we will follow the outlined procedure to calculate the power dissipation in the circular coil.

### Step 1: Identify the given values
- Magnetic field \( B(t) = 0.2t - 0.05t^2 \) (in tesla)
- Number of turns \( N = 25 \)
- Radius of the coil \( r = 1.8 \, \text{cm} = 0.018 \, \text{m} \)
- Resistance \( R = 1.5 \, \Omega \)
- Time \( t = 3 \, \text{s} \)

### Step 2: Calculate the area of the circular coil
The area \( A \) of the circular coil can be calculated using the formula:
\[ A = \pi r^2 \]
Substituting the value of \( r \):
\[ A = \pi (0.018)^2 \approx \pi (0.000324) \approx 0.001018 \, \text{m}^2 \]

### Step 3: Calculate the magnetic flux \( \Phi \)
The magnetic flux \( \Phi \) through the coil is given by:
\[ \Phi = N \cdot B(t) \cdot A \]
Substituting the values:
\[ \Phi = 25 \cdot (0.2t - 0.05t^2) \cdot 0.001018 \]
At \( t = 3 \, \text{s} \):
\[ B(3) = 0.2(3) - 0.05(3^2) = 0.6 - 0.45 = 0.15 \, \text{T} \]
Now substituting \( B(3) \) into the flux equation:
\[ \Phi = 25 \cdot 0.15 \cdot 0.001018 \approx 0.003795 \, \text{Wb} \]

### Step 4: Calculate the induced EMF (\( \mathcal{E} \))
The induced EMF can be calculated using Faraday's law:
\[ \mathcal{E} = -\frac{d\Phi}{dt} \]
To find \( \frac{d\Phi}{dt} \), we need to differentiate \( \Phi \):
\[ \Phi(t) = 25 \cdot (0.2t - 0.05t^2) \cdot 0.001018 \]
Differentiating:
\[ \frac{d\Phi}{dt} = 25 \cdot 0.001018 \cdot (0.2 - 0.1t) \]
At \( t = 3 \):
\[ \frac{d\Phi}{dt} = 25 \cdot 0.001018 \cdot (0.2 - 0.3) = 25 \cdot 0.001018 \cdot (-0.1) \approx -0.0002545 \, \text{Wb/s} \]
Thus, the induced EMF is:
\[ \mathcal{E} = -(-0.0002545) \approx 0.0002545 \, \text{V} \]

### Step 5: Calculate the current \( I \)
Using Ohm's law:
\[ I = \frac{\mathcal{E}}{R} \]
Substituting the values:
\[ I = \frac{0.0002545}{1.5} \approx 0.0001697 \, \text{A} \approx 0.1697 \, \text{mA} \]

### Step 6: Calculate the power \( P \)
The power dissipated in the coil can be calculated using:
\[ P = I^2 R \]
Substituting the values:
\[ P = (0.0001697)^2 \cdot 1.5 \approx 0.0000000287 \cdot 1.5 \approx 0.00000004305 \, \text{W} \approx 4.305 \, \mu W \]

### Final Answer
The power dissipation at \( t = 3 \, \text{s} \) is approximately \( 4.305 \, \mu W \).

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