Two very long straight parallel wires, parallel to y-axis, carry currents 41 and I, along+y direction and-y direaction respectively. The wires are passes through the x-axis at the points `(d, 0,0)` and `(-d, 0,0)` respectively. The graph of magnetic field z-component as one moves along the x-axis from `x = - d` to `x = +d`, is best given by

Options

A.
B.
C.
D.

Explanation

To solve the problem regarding the magnetic field due to two long straight parallel wires, we can follow these steps:

### Step 1: Understand the Configuration
We have two long straight parallel wires:
- Wire 1 carries a current of \(4I\) in the +y direction and is located at the point \((d, 0, 0)\).
- Wire 2 carries a current of \(I\) in the -y direction and is located at the point \((-d, 0, 0)\).

### Step 2: Determine the Magnetic Field Contributions
Using the right-hand rule and the Biot-Savart law, we can find the direction of the magnetic field produced by each wire at a point along the x-axis.

1. **Magnetic Field due to Wire 1 (at \(x = d\))**:
- The magnetic field \(B_1\) at a point \(x\) (where \(x > d\)) due to Wire 1 is directed out of the page (in the +z direction).
- The magnitude of the magnetic field is given by:
\[ B_1 = \frac{\mu_0 (4I)}{2\pi r_1} \]
where \(r_1 = x - d\).

2. **Magnetic Field due to Wire 2 (at \(x = -d\))**:
- The magnetic field \(B_2\) at a point \(x\) (where \(x < -d\)) due to Wire 2 is also directed out of the page (in the +z direction).
- The magnitude of the magnetic field is given by:
\[ B_2 = \frac{\mu_0 I}{2\pi r_2} \]
where \(r_2 = x + d\).

### Step 3: Combine the Magnetic Fields
For \(x\) between \(-d\) and \(d\):
- The magnetic field contributions from both wires will add up since they are in the same direction (both in the +z direction).
- The total magnetic field \(B\) at a point \(x\) is:
\[ B = B_1 + B_2 = \frac{\mu_0 (4I)}{2\pi (d - x)} + \frac{\mu_0 I}{2\pi (d + x)} \]

### Step 4: Analyze the Behavior of the Magnetic Field
- As \(x\) approaches \(-d\), \(B_2\) becomes very large (approaches infinity) because \(r_2\) approaches zero.
- As \(x\) approaches \(d\), \(B_1\) becomes very large (also approaches infinity).
- There will be a point between \(-d\) and \(d\) where the magnetic field is at a minimum due to the opposing effects of the two currents.

### Step 5: Find the Minimum Magnetic Field
To find the point where the magnetic field is minimum, we can differentiate \(B\) with respect to \(x\) and set it to zero:
\[ \frac{dB}{dx} = 0 \]
This will give us the position where the magnetic field is minimized.

### Step 6: Solve for the Minimum Point
After differentiating and simplifying, we find that the magnetic field is minimized at:
\[ x = \frac{2d}{3} \]
This point is closer to Wire 2 than to Wire 1, indicating that the magnetic field strength will be lower there.

### Step 7: Graph the Magnetic Field
The graph of the magnetic field \(B\) as a function of \(x\) will show:
- Very high values (approaching infinity) at \(x = -d\) and \(x = d\).
- A minimum value at \(x = \frac{2d}{3}\).

### Conclusion
The best representation of the magnetic field's z-component as one moves along the x-axis from \(x = -d\) to \(x = d\) is a graph that shows the magnetic field increasing to infinity at both ends and having a minimum at \(x = \frac{2d}{3}\).