The magnetic field at the origin due to a current element I `(vec dl ) placed at a positon (vec r) is`

Options

A. `(((mu_0 i)/(4pi))` `(vec (dl) xx vec (r))/r^3)`
B. `(-((mu_0)i/(4 pi) (vec(r) xx vec(dl) / r^3)))`
C. `((mu_0)i/(4 pi) (vec(r) xx vec(dl) / r^3))`
D. `(-((mu_0)i/(4pi)(vec (dl) xx vec (r))/r^3))`

Answer

To find the magnetic field at the origin due to a current element \( I \, d\vec{l} \) placed at a position \( \vec{r} \), we can use the Biot-Savart law. Here’s a step-by-step solution:

### Step 1: Understand the Biot-Savart Law
The Biot-Savart law states that the magnetic field \( \vec{B} \) at a point in space due to a current element is given by the formula:
\[ \vec{B} = \frac{\mu_0}{4\pi} \frac{I \, d\vec{l} \times \vec{r}}{|\vec{r}|^3} \]
where:
- \( \mu_0 \) is the permeability of free space,
- \( I \) is the current,
- \( d\vec{l} \) is the current element vector,
- \( \vec{r} \) is the position vector from the current element to the point where the magnetic field is being calculated.

### Step 2: Define the Position Vector
In this problem, we need to find the magnetic field at the origin (0,0) due to the current element located at position \( \vec{r} \). The position vector from the current element to the origin is:
\[ \vec{r}_{origin} = -\vec{r} \]
This means we need to consider the negative of the position vector \( \vec{r} \) when applying the Biot-Savart law.

### Step 3: Substitute into the Biot-Savart Law
Substituting \( \vec{r}_{origin} \) into the Biot-Savart law, we have:
\[ \vec{B}_{origin} = \frac{\mu_0}{4\pi} \frac{I \, d\vec{l} \times (-\vec{r})}{|-\vec{r}|^3} \]
Since the magnitude of \( -\vec{r} \) is the same as \( \vec{r} \), we can simplify this to:
\[ \vec{B}_{origin} = -\frac{\mu_0}{4\pi} \frac{I \, d\vec{l} \times \vec{r}}{|\vec{r}|^3} \]

### Step 4: Final Expression
Thus, the magnetic field at the origin due to the current element \( I \, d\vec{l} \) placed at position \( \vec{r} \) is given by:
\[ \vec{B}_{origin} = -\frac{\mu_0 I}{4\pi} \frac{d\vec{l} \times \vec{r}}{|\vec{r}|^3} \]

### Summary
The magnetic field at the origin due to the current element can be expressed as:
\[ \vec{B}_{origin} = -\frac{\mu_0 I}{4\pi} \frac{d\vec{l} \times \vec{r}}{|\vec{r}|^3} \]