Question

The magnetic field at the origin due to a current element \(\overrightarrow{idl}\) placed at a point with vector position \(\overrightarrow{r}\) is

• A. \(\frac{\mu_0 i}{4\pi} \frac{\vec{dl} \times \vec{r}} {{r^3}}\)
• B. \(\frac{\mu_0 i}{4\pi}\frac{\vec{r} \times \vec{dl}}{r^3}\)
• C. \(\frac{\mu_0 i}{4\pi} \frac{\vec{dl} \times \vec{r}} {{r^2}}\)
• D. \(\frac{\mu_0 i}{4\pi}\frac{\vec{r} \times \vec{dl}}{r^2}\)

Answer 1

The Correct Option is A

The magnetic field \(\vec{B}\) due to a current element \(i \, d\vec{l}\) at a point with position vector \(\vec{r}\) is given by the Biot-Savart law: \[ \vec{B} = \frac{\mu_0 i \, d\vec{l} \times \vec{r}}{4 \pi r^3} \] Where:
\(\mu_0\) is the permeability of free space,
\(i\) is the current,
\(d\vec{l}\) is the current element,
\(\vec{r}\) is the position vector,
\(r\) is the distance between the current element and the point where the magnetic field is being calculated.
Thus, the magnetic field at the origin due to the current element is given by option (A).

The correct option is (A):\(\frac{\mu_0 i}{4\pi} \frac{\vec{dl} \times \vec{r}} {{r^3}}\)

Answer 2

The magnetic field \( \mathbf{B} \) due to a current element \( \mathbf{idl} \) at a point with position vector \( \mathbf{r} \) from the origin is given by the Biot-Savart Law: \[ \mathbf{B} = \frac{\mu_0}{4 \pi} \frac{i \, \mathbf{dl} \times \mathbf{r}}{r^3} \] Where: 
\( \mu_0 \) is the permeability of free space,
\( i \) is the current in the element,
\( \mathbf{dl} \) is the vector representing the current element,
\( \mathbf{r} \) is the position vector from the origin to the point where the magnetic field is being calculated, 
\( r \) is the magnitude of \( \mathbf{r} \).

The expression above shows that the magnetic field at the origin due to a current element is proportional to the cross product of the position vector \( \mathbf{r} \) and the current element \( \mathbf{dl} \), and inversely proportional to the cube of the distance \( r^3 \). Thus, the correct expression is: \[ \mathbf{B} = \frac{\mu_0 i \mathbf{dl} \times \mathbf{r}}{4 \pi r^3} \] Therefore, the correct answer is (A).