To find the current around the washer, we will follow these steps:
### Step 1: Calculate the Area of the Washer
The area \( A \) of the washer can be calculated using the formula for the area of a ring:
\[
A = \pi (R^2 - r^2)
\]
where \( R \) is the outer radius and \( r \) is the inner radius.
Given:
- Inner radius \( r = 1 \, \text{cm} = 0.01 \, \text{m} \)
- Outer radius \( R = 3 \, \text{cm} = 0.03 \, \text{m} \)
Calculating the area:
\[
A = \pi \left((0.03)^2 - (0.01)^2\right) = \pi \left(0.0009 - 0.0001\right) = \pi \times 0.0008 = 0.0008\pi \, \text{m}^2
\]
### Step 2: Calculate the Change in Magnetic Flux
The magnetic field \( B \) is given as \( B = 2t \, \text{T} \). The magnetic flux \( \Phi \) through the washer is given by:
\[
\Phi = B \cdot A
\]
Thus,
\[
\Phi = (2t) \cdot (0.0008\pi) = 0.0016\pi t \, \text{Wb}
\]
### Step 3: Calculate the Induced EMF
The induced EMF \( \mathcal{E} \) can be calculated using Faraday's law of electromagnetic induction:
\[
\mathcal{E} = -\frac{d\Phi}{dt}
\]
Differentiating the flux:
\[
\mathcal{E} = -\frac{d}{dt}(0.0016\pi t) = -0.0016\pi \, \text{V}
\]
Since we are interested in the magnitude, we take:
\[
\mathcal{E} = 0.0016\pi \, \text{V}
\]
### Step 4: Calculate the Resistance of the Washer
The resistance \( R \) of the washer can be calculated using the formula:
\[
R = \frac{\rho L}{A}
\]
where:
- \( \rho = 10^{-7} \, \Omega \cdot \text{m} \) (resistivity)
- \( L \) (the average circumference of the washer) can be calculated as \( L = 2\pi \times \text{average radius} \).
The average radius \( r_{avg} \) is:
\[
r_{avg} = \frac{R + r}{2} = \frac{0.03 + 0.01}{2} = 0.02 \, \text{m}
\]
Thus,
\[
L = 2\pi(0.02) = 0.04\pi \, \text{m}
\]
Now substituting into the resistance formula:
\[
R = \frac{(10^{-7})(0.04\pi)}{0.0008\pi} = \frac{10^{-7} \cdot 0.04}{0.0008} = \frac{10^{-7} \cdot 0.04}{8 \times 10^{-4}} = 5 \times 10^{-6} \, \Omega
\]
### Step 5: Calculate the Current
Using Ohm's law:
\[
I = \frac{\mathcal{E}}{R}
\]
Substituting the values:
\[
I = \frac{0.0016\pi}{5 \times 10^{-6}} = \frac{0.0016 \times 3.14}{5 \times 10^{-6}} \approx \frac{0.005024}{5 \times 10^{-6}} = 1004.8 \, \text{A}
\]
### Final Answer
The current around the washer is approximately \( 1004.8 \, \text{A} \).
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