Question

Biot-Savart law indicates that an electron moving with a velocity \( \vec{V} \) produces a magnetic field \( \vec{B} \) around it such that

• A. \( \vec{B} \) is parallel to \( \vec{V} \)
• B. \( \vec{B} \) is perpendicular to \( \vec{V} \)
• C. \( \vec{B} \) is anti-parallel to \( \vec{V} \)
• D. \( \vec{B} \) is inclined to \( \vec{V} \) by 45\(^{\circ}\)

Hint:

Remember the "Right-Hand Rule": the magnetic field lines form circles around the path of the moving charge. In any instantaneous cross-section, the field vector at any point will always be in a direction perpendicular to the velocity vector.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks about the orientation of the magnetic field produced by a moving point charge (an electron) relative to its direction of motion.
Step 2: Key Formula or Approach:
The Biot-Savart law for a point charge \( q \) moving with velocity \( \vec{V} \) is given by:
\[ \vec{B} = \frac{\mu_0}{4\pi} \frac{q(\vec{V} \times \hat{r})}{r^2} \] Where \( \hat{r} \) is the unit vector from the charge to the point of observation.
Step 3: Detailed Explanation:
According to the cross-product property in the formula \( \vec{B} \propto \vec{V} \times \hat{r} \), the resultant vector \( \vec{B} \) is always perpendicular to the plane containing the velocity vector \( \vec{V} \) and the position vector \( \vec{r} \).
This fundamental property implies that the magnetic field \( \vec{B} \) at any point in space is always perpendicular to the velocity vector \( \vec{V} \) of the moving charge.
Step 4: Final Answer:
Therefore, \( \vec{B} \) is always perpendicular to \( \vec{V} \), which corresponds to option (2).