Biot-Savart law indicates that the moving electrons (velocity `vecv` ) produce a magnetic field `vecB` such that

Options

A. `vec(B) _|_vec(v)`
B. `vec(B) || vec(v)`
C. it obeys inverse cube law.
D. it is along the line joining the electron and point of observation.

Explanation

To solve the question regarding the Biot-Savart law and the characteristics of the magnetic field produced by moving electrons, we can break it down step by step.

### Step-by-Step Solution:

1. **Understanding Biot-Savart Law**: The Biot-Savart law describes how moving charges produce a magnetic field. The law can be mathematically expressed as:
\[ \vec{B} = \frac{\mu_0}{4\pi} \frac{q \vec{v} \times \vec{r}}{r^3} \]
where:
- \(\vec{B}\) is the magnetic field,
- \(\mu_0\) is the permeability of free space,
- \(q\) is the charge,
- \(\vec{v}\) is the velocity of the charge,
- \(\vec{r}\) is the position vector from the charge to the point of observation,
- \(r\) is the magnitude of \(\vec{r}\).

2. **Direction of the Magnetic Field**: The direction of the magnetic field \(\vec{B}\) is given by the right-hand rule, which states that if you point your thumb in the direction of \(\vec{v}\) (the velocity of the charge) and curl your fingers towards \(\vec{r}\) (the position vector), your palm will point in the direction of \(\vec{B}\). This indicates that \(\vec{B}\) is perpendicular to both \(\vec{v}\) and \(\vec{r}\).

3. **Magnitude of the Magnetic Field**: The magnitude of the magnetic field can be simplified from the Biot-Savart law:
\[ |\vec{B}| = \frac{\mu_0}{4\pi} \frac{q v \sin \theta}{r^2} \]
where \(\theta\) is the angle between \(\vec{v}\) and \(\vec{r}\). This shows that the magnetic field strength is inversely proportional to the square of the distance \(r\) from the charge, not the cube.

4. **Evaluating the Options**:
- **Option A**: \(\vec{B}\) is perpendicular to \(\vec{v}\) (Correct).
- **Option B**: \(\vec{B}\) is parallel to \(\vec{v}\) (Incorrect).
- **Option C**: It obeys inverse cube law (Incorrect; it obeys inverse square law).
- **Option D**: It is along the line joining the electron and the point of observation (Incorrect).

5. **Conclusion**: The correct answer is that the magnetic field \(\vec{B}\) produced by moving electrons is perpendicular to the velocity \(\vec{v}\).

### Final Answer:
The correct option is **A**: \(\vec{B}\) is perpendicular to \(\vec{v}\).