Question

A straight conductor lies along x-axis and carries a current of 2 A along +x direction. The magnetic field at a point (0, 40 cm, 0) due to 1 cm length of conductor centered at the origin points along.

• A. y-axis
• B. –y-axis
• C. z-axis
• D. –z-axis

Hint:

For cross products, consistently picture the Right-Hand Rule: If your index finger points to the current ($\hat{i}$) and your middle finger points to the location ($\hat{j}$), your thumb automatically points to the magnetic field ($\hat{k}$).

Answer 1

The Correct Option is C

Concept:
• The Biot-Savart Law mathematically calculates the exact magnetic field generated by an infinitesimally small current-carrying wire segment at any specific point in surrounding space.

• The law is officially stated in its rigorous vector form as: $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I(d\vec{l} \times \vec{r})}{r^3}$.

• The definitive direction of the resulting magnetic field $d\vec{B}$ is strictly dictated by the mathematical cross product of the current element vector $d\vec{l}$ and the position vector $\vec{r}$.

• The cross product inherently guarantees that the resulting magnetic field vector is definitively perpendicular to both the current element vector and the position vector, adhering strictly to the right-hand rule.

Step 1: Identify the fundamental vector components from the problem
The conductive wire carries an active current flowing strictly along the positive x-axis.
Therefore, the infinitesimal current element vector $d\vec{l}$ points squarely in the $+\hat{i}$ direction:
\[ d\vec{l} = dx \hat{i} \]
The target observation point is located specifically at the spatial coordinates $(0, 40 \text{ cm}, 0)$.
This point clearly lies entirely on the positive y-axis.
Therefore, the position vector $\vec{r}$ extending directly from the origin to the observation point firmly points in the $+\hat{j}$ direction:
\[ \vec{r} = 40 \hat{j} \text{ cm} \]

Step 2: Execute the required vector cross product
To confidently ascertain the specific direction of the generated magnetic field $d\vec{B}$, we must carefully evaluate the core cross product $(d\vec{l} \times \vec{r})$.
We substitute the identified unit vector directions directly into the cross product:
\[ \text{Direction of } d\vec{B} \propto (\hat{i} \times \hat{j}) \]
According to the universally established standard cyclic rules for orthogonal Cartesian unit vectors:
\[ \hat{i} \times \hat{j} = \hat{k} \]

Step 3: Interpret the mathematical result physically
The final unit vector $\hat{k}$ definitively represents the positive z-axis in a standard 3D coordinate system.
Therefore, the infinitesimally small magnetic field generated strictly by that tiny 1 cm central segment points straight out along the positive z-axis.

Step 4: Conclusion
The generated magnetic field points robustly along the z-axis, which flawlessly matches option (C).