An electric current is flowing through a long cylindrical conductor with radius a = 0.55 m. The current density J = 9.5 A/m2 is uniform in the cylinder. In this problem we consider an imaginary cylinder with radius r around the axis AB. For r=2a,calculate the value B in Tesla

To find the magnetic field \( B \) at a radius \( r = 2a \) from the axis of a long cylindrical conductor with a uniform current density, we can apply Ampère's law.

### Step-by-Step Solution:

1. **Given Information**:
   - Radius of the conductor, \( a = 0.55 \, \text{m} \)
   - Current density, \( J = 9.5 \, \text{A/m}^2 \)
   - Radius of the imaginary cylinder, \( r = 2a = 2 \times 0.55 = 1.1 \, \text{m} \)

2. **Calculate the Total Current \( I \)**:
   Since the current density \( J \) is uniform, the total current \( I \) flowing through the conductor can be calculated as:
   \[
   I = J \cdot A
   \]
   where \( A \) is the cross-sectional area of the conductor:
   \[
   A = \pi a^2
   \]
   Thus,
   \[
   I = J \cdot \pi a^2 = 9.5 \times \pi \times (0.55)^2
   \]

3. **Substitute and Calculate \( I \)**:
   \[
   I = 9.5 \times \pi \times 0.3025 \approx 9.019 \, \text{A}
   \]

4. **Apply Ampère's Law to Find \( B \)**:
   Ampère's Law states:
   \[
   \oint B \, \text{d}l = \mu_0 I_{\text{enc}}
   \]
   where \( I_{\text{enc}} \) is the current enclosed within the path of radius \( r \), and \( \mu_0 \) is the permeability of free space (\( \mu_0 = 4\pi \times 10^{-7} \, \text{T m/A} \)).

   For \( r = 2a \), which is outside the conductor, the entire current \( I \) is enclosed. Since the magnetic field \( B \) is constant over the circular path at radius \( r \), we can simplify this to:
   \[
   B \cdot (2\pi r) = \mu_0 I
   \]
   Solving for \( B \):
   \[
   B = \frac{\mu_0 I}{2\pi r}
   \]

5. **Substitute and Calculate \( B \)**:
   \[
   B = \frac{4\pi \times 10^{-7} \times 9.019}{2\pi \times 1.1}
   \]
   Simplify by canceling \( \pi \) and calculating:
   \[
   B \approx \frac{4 \times 10^{-7} \times 9.019}{2 \times 1.1} \approx 1.637 \times 10^{-6} \, \text{T}
   \]

### Final Answer:
\[
B \approx 1.64 \times 10^{-6} \, \text{T}
\]

Would you like more details, or do you have any further questions?

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### Related Questions:
1. How does the magnetic field vary with distance inside the conductor?
2. What happens to the magnetic field at points beyond \( r = 2a \)?
3. How would the result change if the current density was not uniform?
4. How does Ampère's law apply to conductors with varying cross-sections?
5. How would the calculation change if the conductor were hollow?

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### Tip:
For problems involving current distributions, always check if the current is uniformly distributed, as it simplifies calculations significantly with Ampère's law.

An electric current is flowing through a long cylindrical conductor with radius a = 0.55 m. The current density J = 9.5 A/m² is uniform in the cylinder. In this problem we consider an imaginary cylinder with radius r around the axis AB. For r=2a, calculate the value B in Tesla.

Calculate the magnetic field at a distance r = 2a from the axis of a cylindrical conductor with uniform current density. Given radius a = 0.55 m and current density J = 9.5 A/m², apply Ampère's Law for a step-by-step solution to find B in Tesla.

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