A long, straight wire carries a current along the z-axis. One can find two points in the x-y plane such that

Options

A. the magnetic fields are equal
B. the directions of the 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 problem, we need to analyze the magnetic field produced by a long straight wire carrying a current along the z-axis. The magnetic field at any point in the x-y plane due to this wire can be calculated using the formula:

\[ B = \frac{\mu_0 I}{2\pi d} \]

where:
- \( B \) is the magnetic field,
- \( \mu_0 \) is the permeability of free space,
- \( I \) is the current flowing through the wire,
- \( d \) is the perpendicular distance from the wire to the point where the magnetic field is being calculated.

### Step-by-Step Solution:

1. **Identify the Wire and Points**:
- The wire is oriented along the z-axis.
- We need to find two points in the x-y plane where the magnetic fields can be compared.

2. **Magnetic Field Calculation**:
- For a point at distance \( d_1 \) from the wire, the magnetic field \( B_1 \) is given by:
\[ B_1 = \frac{\mu_0 I}{2\pi d_1} \]
- For another point at distance \( d_2 \) from the wire, the magnetic field \( B_2 \) is given by:
\[ B_2 = \frac{\mu_0 I}{2\pi d_2} \]

3. **Direction of the Magnetic Field**:
- Using the right-hand rule, the direction of the magnetic field will depend on the position of the points relative to the wire.
- If one point is above the wire and the other is below, the magnetic fields at these points will be in opposite directions.

4. **Magnitude of the Magnetic Fields**:
- If \( d_1 = d_2 \), then \( B_1 = B_2 \).
- If \( d_1 \neq d_2 \), the magnitudes will differ, but we can find points where the magnitudes are equal by setting \( d_1 = d \) and \( d_2 = -d \) (i.e., symmetrical points).

5. **Conclusion**:
- The magnetic fields at the two points can be equal in magnitude but opposite in direction if they are equidistant from the wire on opposite sides.
- Therefore, the correct options based on the analysis are:
- The magnitudes of the magnetic fields are equal.
- The directions of the magnetic fields are opposite.

### Final Answer:
- The correct options are:
- **C**: The magnitudes of the magnetic fields are equal.
- **D**: The field at one point is opposite to that at the other point.