The electric field outside a charged long straight wire is given by `E = (1000)/(r ) V m^(-1)`, and is directed outwards. What is the sign of the charge on the wire ? If two points `A` and `B` are situated such that `r_(A) = 0.2 m` and `r_(B) = 0.4 m`, find the value of `(V_(B) - V_(A))`.

Explanation

To solve the problem step by step, we will first determine the sign of the charge on the wire and then calculate the potential difference \( V_B - V_A \) between points A and B.

### Step 1: Determine the sign of the charge on the wire

Given that the electric field \( E \) outside the charged long straight wire is given by:

\[ E = \frac{1000}{r} \, \text{V/m} \]

and it is directed outwards. The direction of the electric field indicates the nature of the charge. If the electric field is directed outward from the wire, it means that the wire has a positive charge.

**Conclusion for Step 1:**
The charge on the wire is **positive**.

### Step 2: Calculate the potential difference \( V_B - V_A \)

The potential difference between two points in an electric field can be calculated using the formula:

\[ V_B - V_A = -\int_{r_A}^{r_B} E \, dr \]

Given that:

- \( r_A = 0.2 \, \text{m} \)
- \( r_B = 0.4 \, \text{m} \)

Substituting the expression for \( E \):

\[ V_B - V_A = -\int_{0.2}^{0.4} \frac{1000}{r} \, dr \]

### Step 3: Perform the integration

The integral of \( \frac{1}{r} \) is:

\[ \int \frac{1}{r} \, dr = \ln r \]

Thus, we have:

\[ V_B - V_A = -1000 \left[ \ln r \right]_{0.2}^{0.4} \]

Calculating the definite integral:

\[ = -1000 \left( \ln(0.4) - \ln(0.2) \right) \]

Using the property of logarithms:

\[ \ln(0.4) - \ln(0.2) = \ln\left(\frac{0.4}{0.2}\right) = \ln(2) \]

So we can rewrite the potential difference as:

\[ V_B - V_A = -1000 \ln(2) \]

### Step 4: Substitute the value of \( \ln(2) \)

Using the approximate value \( \ln(2) \approx 0.693 \):

\[ V_B - V_A = -1000 \times 0.693 = -693 \, \text{V} \]

### Final Answer

Thus, the potential difference \( V_B - V_A \) is:

\[ V_B - V_A = -693 \, \text{V} \]