What is the line charge density on a long wire if a 6.7- μg
  particle carrying 2.2  nC
  describes a circular orbit about the wire with speed 320  m/s
 ?
Express your answer in microcoulombs per meter.

To determine the line charge density (λ) on a long wire given that a charged particle describes a circular orbit around it, we'll apply the principles of electrostatics and centripetal force.

### Given:
- Mass of the particle, \( m = 6.7 \, \mu\text{g} = 6.7 \times 10^{-9} \, \text{kg} \)
- Charge of the particle, \( q = 2.2 \, \text{nC} = 2.2 \times 10^{-9} \, \text{C} \)
- Speed of the particle, \( v = 320 \, \text{m/s} \)

### Formula:
The centripetal force required to keep the particle in circular motion is provided by the electrostatic force between the charged particle and the line charge of the wire.

1. **Centripetal force:**
   \[
   F_{\text{centripetal}} = \frac{mv^2}{r}
   \]
   where \( r \) is the radius of the circular orbit.

2. **Electrostatic force:**
   The electric field \( E \) due to an infinite line charge is given by:
   \[
   E = \frac{\lambda}{2 \pi \epsilon_0 r}
   \]
   where \( \lambda \) is the line charge density, and \( \epsilon_0 \) is the permittivity of free space (\( \epsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2/\text{N}\cdot\text{m}^2 \)).

   The force on the particle is:
   \[
   F_{\text{electrostatic}} = qE = q \frac{\lambda}{2 \pi \epsilon_0 r}
   \]

Since the electrostatic force provides the centripetal force, we equate these:
\[
\frac{mv^2}{r} = q \frac{\lambda}{2 \pi \epsilon_0 r}
\]

The radius \( r \) cancels out from both sides:
\[
\lambda = \frac{2 \pi \epsilon_0 mv^2}{q}
\]

### Plug in the values:
\[
\lambda = \frac{2 \pi (8.85 \times 10^{-12} \, \text{C}^2/\text{N}\cdot\text{m}^2) \times (6.7 \times 10^{-9} \, \text{kg}) \times (320 \, \text{m/s})^2}{2.2 \times 10^{-9} \, \text{C}}
\]

### Calculate:

\[
\lambda \approx \frac{2 \pi \times 8.85 \times 10^{-12} \times 6.7 \times 10^{-9} \times 102400}{2.2 \times 10^{-9}}
\]

\[
\lambda \approx \frac{2 \times 3.1416 \times 8.85 \times 6.7 \times 102400 \times 10^{-21}}{2.2 \times 10^{-9}}
\]

\[
\lambda \approx \frac{3.706 \times 10^{-12}}{2.2 \times 10^{-9}}
\]

\[
\lambda \approx 1.684 \times 10^{-3} \, \text{C/m}
\]

Finally, convert to microcoulombs per meter:
\[
\lambda \approx 1.684 \, \mu\text{C/m}
\]

### Answer:
The line charge density on the wire is approximately **1.684 μC/m**.

Would you like to know more details or have any questions?

### Related Questions:
1. How would the line charge density change if the speed of the particle doubled?
2. What would be the impact on the orbit if the mass of the particle increased?
3. How does the radius of the orbit relate to the line charge density?
4. What assumptions are made in using the formula for an infinite line charge?
5. How would the line charge density be affected if the charge of the particle was negative?

### Tip:
Always check the units when plugging values into formulas to ensure consistency and correct final units.

Learn how to determine the line charge density on a long wire using principles of electrostatics and centripetal force. Given a particle's mass, charge, and speed in circular orbit, calculate the line charge density in microcoulombs per meter (μC/m).

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