Two infinitely long, thin, insulated, straight wires lie in the x-y plane along the x- and y- axis respectively. Each wire carries a current I, respectively in the positive x-direction and positive y-direction. The magnetic field will be zero at all points on the straight line:

Options

A. y = x
B. y = -x
C. y = x-1
D. y = -x+1

Answer

To solve the problem, we need to find the condition under which the magnetic field produced by two infinitely long, straight wires carrying currents in perpendicular directions (one along the x-axis and the other along the y-axis) is zero at all points on a straight line.

### Step-by-step Solution:

1. **Identify the Configuration**:
- We have two wires: one along the x-axis carrying current \( I \) in the positive x-direction, and the other along the y-axis carrying current \( I \) in the positive y-direction.

2. **Magnetic Field due to a Long Straight Wire**:
- The magnetic field \( B \) at a distance \( r \) from a long straight wire carrying current \( I \) is given by the formula:
\[ B = \frac{\mu_0 I}{2 \pi r} \]
- The direction of the magnetic field can be determined using the right-hand rule.

3. **Calculate Magnetic Field at a Point (x, y)**:
- Consider a point \( P(x, y) \) in the x-y plane.
- The distance from the point \( P \) to the wire along the x-axis is \( y \) (perpendicular distance).
- The distance from the point \( P \) to the wire along the y-axis is \( x \) (perpendicular distance).

4. **Magnetic Field from the Wire along the x-axis**:
- The magnetic field \( B_x \) at point \( P \) due to the wire along the x-axis (current in positive x-direction) is directed out of the page (upward):
\[ B_x = \frac{\mu_0 I}{2 \pi y} \]

5. **Magnetic Field from the Wire along the y-axis**:
- The magnetic field \( B_y \) at point \( P \) due to the wire along the y-axis (current in positive y-direction) is directed into the page (downward):
\[ B_y = \frac{\mu_0 I}{2 \pi x} \]

6. **Setting the Magnetic Fields Equal**:
- For the total magnetic field to be zero at point \( P \), the magnitudes of \( B_x \) and \( B_y \) must be equal:
\[ B_x = B_y \]
- Therefore, we have:
\[ \frac{\mu_0 I}{2 \pi y} = \frac{\mu_0 I}{2 \pi x} \]

7. **Simplifying the Equation**:
- Canceling common terms:
\[ \frac{1}{y} = \frac{1}{x} \]
- This implies:
\[ x = y \]

8. **Conclusion**:
- The magnetic field will be zero at all points on the straight line where \( x = y \). This line is at a 45-degree angle to both axes in the first quadrant.

### Final Answer:
The magnetic field will be zero at all points on the straight line where \( x = y \).