Read the following paragraph and answer the question:
A magnetic field can be produced by moving charges or electric currents. The basic equation governing the magnetic field due to a current distribution is the Biot-Savart law.
Finding the magnetic field resulting from a current distribution involves the vector product and is inherently a calculus problem when the distance from the current to the field point is continuously changing.
According to this law, the magnetic field at a point due to a current element of length `vec(dl)` carrying current I, at a distance r from the element is dB = `(mu_0)/(4pi) (I(dvec1 xx vecr))/(r^3)`.
Biot-Savart law has certain similarities as well as difference with Coulomb's law for electrostatic field e.g., there is an angle dependence in Biot-Savart law which is not present in electrostatic case.
A long straight wire carries a current along the z-axis for any two points in the x-y plane. Which of the following is always false ?

Options

A. The magnetic field are equal
B. The directions of other magnetic fields are the same
C. The magnitudes of the magnetic fields are equal
D. The field at one point is opposite to that at the other point

Answer

To solve the question regarding the magnetic field produced by a long straight wire carrying current along the z-axis, we need to analyze the statements provided and determine which one is always false.

### Step-by-Step Solution:

1. **Understanding the Biot-Savart Law**:
The Biot-Savart Law states that the magnetic field \( dB \) at a point due to a current element \( dL \) carrying current \( I \) is given by:
\[ dB = \frac{\mu_0}{4\pi} \frac{I \, dL \times \hat{r}}{r^2} \]
where \( \hat{r} \) is the unit vector pointing from the current element to the point of interest, and \( r \) is the distance from the current element to that point.

**Hint**: Recall that the magnetic field is dependent on the current, the length of the current element, and the angle between the current element and the position vector.

2. **Magnetic Field Direction**:
The direction of the magnetic field produced by a straight current-carrying wire can be determined using the right-hand rule. If the current flows in the positive z-direction, the magnetic field will circulate around the wire in the xy-plane.

**Hint**: Use the right-hand rule to visualize how the magnetic field circulates around the wire.

3. **Analyzing Points in the XY Plane**:
For any two points in the xy-plane, if they are equidistant from the wire but on opposite sides, the magnitudes of the magnetic fields at these points will be the same, but their directions will be opposite.

**Hint**: Consider the symmetry of the magnetic field around the wire. Points at the same distance but on opposite sides will have fields that are equal in magnitude but opposite in direction.

4. **Evaluating the Statements**:
- **Statement 1**: "The magnetic field at two points is the same."
- This is false if the points are on opposite sides of the wire.
- **Statement 2**: "The direction of the magnetic fields at two points is the same."
- This is false if the points are on opposite sides of the wire.
- **Statement 3**: "The magnitudes of the magnetic fields are equal."
- This is true if the points are equidistant from the wire.
- **Statement 4**: "The field at one point is opposite to that at the other point."
- This is true if the points are on opposite sides of the wire.

**Hint**: Compare the statements to the behavior of the magnetic field around the wire. Identify which statement contradicts the established principles.

5. **Conclusion**:
The statement that is always false is **Statement 1**: "The magnetic field at two points is the same." This is incorrect because the direction of the magnetic field will differ if the points are on opposite sides of the wire.

**Final Answer**: The statement that is always false is: "The magnetic field at two points is the same."