A straight wire 10 m
 long carries 28 μC
 distributed uniformly over its length.  Find the electric field strength 470  m
  from the wire.
Note: make suitable approximation. Express your answer in newtons per coulomb

To find the electric field strength \( E \) at a distance \( r \) from a uniformly charged straight wire, we can use the formula derived from Gauss's law for an infinitely long wire. Given that the length of the wire is much smaller than the distance at which we're measuring the field, this approximation is reasonable.

The formula for the electric field due to a uniformly charged infinite wire is:

\[
E = \frac{\lambda}{2 \pi \epsilon_0 r}
\]

where:
- \( \lambda \) is the linear charge density (charge per unit length),
- \( \epsilon_0 \) is the permittivity of free space (\( 8.854 \times 10^{-12} \, \text{C}^2/\text{N} \cdot \text{m}^2 \)),
- \( r \) is the distance from the wire.

### Step 1: Calculate the linear charge density \( \lambda \)
Given:
- Total charge \( Q = 28 \, \mu\text{C} = 28 \times 10^{-6} \, \text{C} \),
- Length of the wire \( L = 10 \, \text{m} \).

\[
\lambda = \frac{Q}{L} = \frac{28 \times 10^{-6} \, \text{C}}{10 \, \text{m}} = 2.8 \times 10^{-6} \, \text{C/m}
\]

### Step 2: Calculate the electric field strength \( E \)
Given:
- \( r = 470 \, \text{m} \),
- \( \epsilon_0 = 8.854 \times 10^{-12} \, \text{C}^2/\text{N} \cdot \text{m}^2 \).

\[
E = \frac{\lambda}{2 \pi \epsilon_0 r} = \frac{2.8 \times 10^{-6} \, \text{C/m}}{2 \pi \times 8.854 \times 10^{-12} \, \text{C}^2/\text{N} \cdot \text{m}^2 \times 470 \, \text{m}}
\]

\[
E = \frac{2.8 \times 10^{-6}}{2 \times 3.1416 \times 8.854 \times 10^{-12} \times 470}
\]

\[
E \approx \frac{2.8 \times 10^{-6}}{2.616 \times 10^{-8}} \, \text{N/C}
\]

\[
E \approx 107.1 \, \text{N/C}
\]

So, the electric field strength \( 470 \, \text{m} \) from the wire is approximately \( 107.1 \, \text{N/C} \).

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

Here are five related questions:
1. How does the electric field vary with distance from an infinite line of charge?
2. What would happen to the electric field strength if the wire's length was much longer?
3. How would the electric field change if the charge density on the wire increased?
4. Can this method be applied to find the electric field at a point very close to the wire?
5. How does the permittivity of free space (\( \epsilon_0 \)) affect the electric field?

**Tip:** When using approximations, ensure that the assumptions made (like treating a finite wire as infinite) are valid for the given problem conditions.

Learn how to calculate the electric field strength at a distance of 470 meters from a straight wire carrying 28 microcoulombs uniformly distributed over its length. Detailed solution using Gauss's Law approximation and formulas. Suitable for advanced high school students and anyone interested in physics and electricity.

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