The Biot Savart.s Law in vector form is :

Options

A. `vec(deltaB)=(mu_(0))/(4pi).(dI(vecIxxvecr))/(r^(3))`
B. `vec(deltaB)=(mu_(0))/(4pi).(I(vec(dI)xxvecr))/(r^(2))`
C. `vec(deltaB)=(mu_(0))/(4pi).(I(vecrxxvec (dl)))/(r^(3))`
D. `vec(deltaB)=(mu_(0))/(4pi).(I(vec(dl)xxvecr))/(r)`

Explanation

To derive the Biot-Savart Law in vector form, we start with the basic principles of magnetism and the influence of electric currents on the magnetic field.

### Step-by-Step Solution:

1. **Understanding the Concept**:
The Biot-Savart Law describes how a current-carrying conductor produces a magnetic field in the surrounding space. We consider a small segment of the conductor, referred to as the current element (dl).

2. **Defining the Parameters**:
- Let \( I \) be the current flowing through the conductor.
- Let \( dl \) be the differential length vector of the current element.
- Let \( r \) be the distance from the current element to the point where we want to calculate the magnetic field.
- Let \( \theta \) be the angle between the current element \( dl \) and the radial vector \( \mathbf{r} \) pointing from the current element to the observation point.

3. **Magnitude of the Magnetic Field**:
The magnitude of the magnetic field \( dB \) produced by the current element \( dl \) at a distance \( r \) is given by:
\[ dB \propto \frac{I \, dl \, \sin \theta}{r^2} \]
This indicates that the magnetic field is directly proportional to the current \( I \), the length of the current element \( dl \), and the sine of the angle \( \theta \), while inversely proportional to the square of the distance \( r \).

4. **Vector Formulation**:
To express this in vector form, we introduce the unit vector \( \hat{r} \) which points from the current element to the point of interest. The unit vector can be expressed as:
\[ \hat{r} = \frac{\mathbf{r}}{|\mathbf{r}|} \]
where \( |\mathbf{r}| \) is the magnitude of the vector \( \mathbf{r} \).

5. **Using the Cross Product**:
The direction of the magnetic field produced by the current element can be determined using the right-hand rule, which leads us to use the cross product:
\[ d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{I \, d\mathbf{l} \times \hat{r}}{r^2} \]
where \( \mu_0 \) is the permeability of free space.

6. **Final Formulation**:
Therefore, the complete expression for the Biot-Savart Law in vector form becomes:
\[ d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{I \, d\mathbf{l} \times \mathbf{r}}{|\mathbf{r}|^3} \]

### Conclusion:
The Biot-Savart Law in vector form is:
\[ d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{I \, d\mathbf{l} \times \mathbf{r}}{|\mathbf{r}|^3} \]