Vector form of Biot-savart's law is

Options

A. `dvecB = (mu_(0))/(4pi) ((dvecl xx vecr)/( r))`
B. `dvecB = (mu_(0))/(4pi) ""i^(2) ((dvecl xx vecr)/( r))`
C. `dvecB = (mu_(0))/(4pi) ""i^(2) ((d vecl xx vecr)/( r^(2)))`
D. `dvecB = (mu_(0))/(4pi) ""i((dvecl xx vecr)/(r^(3)))`

Explanation

To derive the vector form of Biot-Savart's law, we can follow these steps:

### Step 1: Understanding the Biot-Savart Law
Biot-Savart Law describes the magnetic field generated by a current-carrying conductor. It states that the magnetic field \( \mathbf{B} \) at a point in space is directly proportional to the current \( I \) and the length element \( d\mathbf{L} \) of the conductor, and inversely proportional to the square of the distance \( r \) from the current element to the point where the magnetic field is being calculated.

### Step 2: Setting Up the Equation
The vector form of Biot-Savart's law can be expressed mathematically as:
\[ \mathbf{B} = \frac{\mu_0}{4\pi} \int \frac{I \, d\mathbf{L} \times \mathbf{r}}{r^3} \]
Where:
- \( \mathbf{B} \) is the magnetic field vector.
- \( \mu_0 \) is the permeability of free space.
- \( I \) is the current flowing through the wire.
- \( d\mathbf{L} \) is the differential length vector of the wire.
- \( \mathbf{r} \) is the position vector from the current element to the point where the magnetic field is being measured.
- \( r \) is the magnitude of the vector \( \mathbf{r} \).

### Step 3: Analyzing the Components
1. **Differential Length Element**: The vector \( d\mathbf{L} \) represents a small segment of the wire carrying current \( I \).
2. **Position Vector**: The vector \( \mathbf{r} \) points from the current element \( d\mathbf{L} \) to the point of interest where the magnetic field is being calculated.
3. **Cross Product**: The term \( d\mathbf{L} \times \mathbf{r} \) indicates that the magnetic field is perpendicular to both the current element and the line connecting the current element to the point of interest.

### Step 4: Finalizing the Expression
Thus, the complete expression for the magnetic field \( \mathbf{B} \) due to a small segment of current-carrying wire can be summarized as:
\[ \mathbf{B} = \frac{\mu_0 I}{4\pi} \frac{d\mathbf{L} \times \mathbf{r}}{r^3} \]

### Step 5: Conclusion
The vector form of Biot-Savart's law is:
\[ \mathbf{B} = \frac{\mu_0 I}{4\pi} \frac{d\mathbf{L} \times \mathbf{r}}{r^3} \]