A long straight wire carries the current along +ve x-direction. Consider four points in space `A(0,1,0), B(0,1,1), C(1,0,1), and D(1,1,1)`. Which of the pairs will have the same magnitude of magnetic field?

Options

A. `A and B`
B. `A and C`
C. `B and C`
D. `B and D`

Explanation

To solve the problem of determining which pairs of points have the same magnitude of magnetic field due to a long straight wire carrying current in the +x-direction, we will follow these steps:

### Step 1: Understand the Magnetic Field due to a Current-Carrying Wire
The magnetic field (B) created by a long straight wire carrying current (I) is given by the formula:
\[ B = \frac{\mu_0 I}{2 \pi R} \]
where \( R \) is the perpendicular distance from the wire to the point where the magnetic field is being calculated, and \( \mu_0 \) is the permeability of free space.

### Step 2: Identify the Coordinates of the Points
The points given in the problem are:
- \( A(0, 1, 0) \)
- \( B(0, 1, 1) \)
- \( C(1, 0, 1) \)
- \( D(1, 1, 1) \)

### Step 3: Calculate the Distance \( R \) for Each Point
1. **For Point A (0, 1, 0)**:
- The distance from the wire (which is along the x-axis) to point A is \( R_A = 1 \) (the y-coordinate).
- Therefore, \( B_A = \frac{\mu_0 I}{2 \pi (1)} = \frac{\mu_0 I}{2 \pi} \).

2. **For Point B (0, 1, 1)**:
- The distance from the wire to point B is calculated using the Pythagorean theorem:
\[ R_B = \sqrt{(0-0)^2 + (1-1)^2 + (0-1)^2} = \sqrt{1^2 + 1^2} = \sqrt{2} \]
- Therefore, \( B_B = \frac{\mu_0 I}{2 \pi (\sqrt{2})} = \frac{\mu_0 I}{2 \pi \sqrt{2}} \).

3. **For Point C (1, 0, 1)**:
- The distance from the wire to point C is:
\[ R_C = \sqrt{(1-0)^2 + (0-1)^2} = \sqrt{1^2 + 1^2} = \sqrt{2} \]
- Therefore, \( B_C = \frac{\mu_0 I}{2 \pi (\sqrt{2})} = \frac{\mu_0 I}{2 \pi \sqrt{2}} \).

4. **For Point D (1, 1, 1)**:
- The distance from the wire to point D is:
\[ R_D = \sqrt{(1-0)^2 + (1-1)^2} = \sqrt{1^2 + 1^2} = \sqrt{2} \]
- Therefore, \( B_D = \frac{\mu_0 I}{2 \pi (\sqrt{2})} = \frac{\mu_0 I}{2 \pi \sqrt{2}} \).

### Step 4: Compare the Magnitudes of the Magnetic Fields
From the calculations:
- \( B_A = \frac{\mu_0 I}{2 \pi} \)
- \( B_B = B_C = B_D = \frac{\mu_0 I}{2 \pi \sqrt{2}} \)

### Conclusion
- The magnetic fields at points B, C, and D are equal.
- The magnetic field at point A is different from those at points B, C, and D.

Thus, the pairs that have the same magnitude of magnetic field are:
- **(B, C)**
- **(B, D)**
- **(C, D)**