Question

A current element produces a magnetic field in the region surrounding it. At any point in space, the magnetic field produced by this current element points in a direction that is a) radial from the current element to the point in space. b) parallel to the current element. c) perpendicular to the current element and to the radial direction.

Short answer

Answer: The correct direction of the magnetic field produced by a current element is perpendicular to the current element and to the radial direction.

Step-by-step solution

Recall the Biot-Savart Law

The Biot-Savart Law is given by the formula: $$ dB = \frac{\mu_0}{4\pi}\frac{I d\vec{l} \times \vec{r}}{r^3} $$ where \(dB\) is the infinitesimal magnetic field, \(\mu_0\) is the permeability of free space, \(I\) is the current passing through the current element, \(d\vec{l}\) is the current element vector, \(\vec{r}\) is the position vector from the current element to the point in space where we want to find the magnetic field, and \(r\) is the distance from the current element to that point.

Analyze the Biot-Savart Law formula

The direction of the magnetic field \(dB\) is determined by the cross product of the current element vector \(d\vec{l}\) and the position vector \(\vec{r}\): $$ dB \propto d\vec{l} \times \vec{r} $$

Check each given option

Now we can analyze each given option to see which one is consistent with the direction provided by the Biot-Savart Law. a) Radial from the current element to the point in space: This option suggests that the magnetic field is in the same direction as the position vector \(\vec{r}\). However, the cross product in the Biot-Savart Law implies that the magnetic field is perpendicular to both \(d\vec{l}\) and \(\vec{r}\). Therefore, this option is not correct. b) Parallel to the current element: This option suggests that the magnetic field is in the same direction as the current element vector \(d\vec{l}\). Again, the cross product in the Biot-Savart Law implies that the magnetic field is perpendicular to both \(d\vec{l}\) and \(\vec{r}\). Hence, this option is also incorrect. c) Perpendicular to the current element and to the radial direction: As we have analyzed, the cross product in the Biot-Savart Law indicates that the magnetic field is indeed perpendicular to both \(d\vec{l}\) and \(\vec{r}\). This option is consistent with the formula, so it is the correct answer.

Conclusion

The direction of the magnetic field produced by a current element is perpendicular to the current element and to the radial direction, which corresponds to option (c).