To solve the problem of finding the current around the washer, we will follow these steps:
### Step 1: Identify the given parameters
- Resistivity of the metal, \( \rho = 10^{-7} \, \Omega \cdot m \)
- Inner radius, \( r_{in} = 1 \, cm = 0.01 \, m \)
- Outer radius, \( r_{out} = 3 \, cm = 0.03 \, m \)
- Thickness, \( t = 1 \, mm = 0.001 \, m \)
- Magnetic field, \( B(t) = 2t \, T \)
### Step 2: Calculate the effective area of the washer
The effective area \( A_{effective} \) of the washer is given by the area of the outer circle minus the area of the inner circle:
\[
A_{effective} = \pi r_{out}^2 - \pi r_{in}^2 = \pi (r_{out}^2 - r_{in}^2)
\]
Substituting the values:
\[
A_{effective} = \pi ((0.03)^2 - (0.01)^2) = \pi (0.0009 - 0.0001) = \pi (0.0008) = 8\pi \times 10^{-4} \, m^2
\]
### Step 3: Calculate the cross-sectional area of the washer
The cross-sectional area \( A_{cross} \) can be calculated as:
\[
A_{cross} = (r_{out} - r_{in}) \times t
\]
Substituting the values:
\[
A_{cross} = (0.03 - 0.01) \times 0.001 = 0.02 \times 0.001 = 2 \times 10^{-5} \, m^2
\]
### Step 4: Calculate the average length of the washer
The average radius \( r_{avg} \) is given by:
\[
r_{avg} = \frac{r_{out} + r_{in}}{2} = \frac{0.03 + 0.01}{2} = 0.02 \, m
\]
The length \( L \) of the washer can be calculated as:
\[
L = 2\pi r_{avg} = 2\pi (0.02) = 0.04\pi \, m
\]
### Step 5: Calculate the induced EMF using Faraday's Law
The magnetic flux \( \Phi \) through the washer is given by:
\[
\Phi = B \cdot A_{effective} = (2t) \cdot (8\pi \times 10^{-4})
\]
Differentiating with respect to time to find the EMF \( \mathcal{E} \):
\[
\mathcal{E} = \frac{d\Phi}{dt} = \frac{d}{dt} \left( (2t) \cdot (8\pi \times 10^{-4}) \right) = 8\pi \times 10^{-4} \cdot 2 = 16\pi \times 10^{-4} \, V
\]
### Step 6: Calculate the resistance of the washer
Using the formula for resistance:
\[
R = \frac{\rho L}{A_{cross}}
\]
Substituting the values:
\[
R = \frac{10^{-7} \cdot (0.04\pi)}{2 \times 10^{-5}} = \frac{4\pi \times 10^{-9}}{2 \times 10^{-5}} = 2\pi \times 10^{-4} \, \Omega
\]
### Step 7: Calculate the current using Ohm's Law
Using Ohm's Law \( I = \frac{V}{R} \):
\[
I = \frac{\mathcal{E}}{R} = \frac{16\pi \times 10^{-4}}{2\pi \times 10^{-4}} = \frac{16}{2} = 8 \, A
\]
### Final Answer
The current around the washer is \( I = 8 \, A \).
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