Repeat the above problem, but with the current in both wires shown in fig. directed into the plane of the figure.
Graph the magnitude of `vecB` at points on the x-axis.

Options

A.
B.
C.
D.

Explanation

To solve the problem of finding the magnetic field \( \vec{B} \) at points on the x-axis due to two parallel wires carrying current into the plane of the figure, we can follow these steps:

### Step 1: Understand the Configuration
We have two long straight wires, both carrying currents \( I \) into the plane of the paper. Let’s denote the distance of each wire from the origin as \( A \). The wires are positioned symmetrically about the origin on the y-axis.

### Step 2: Determine the Magnetic Field Contribution from Each Wire
Using the right-hand rule for magnetic fields around a current-carrying wire, the magnetic field \( \vec{B} \) at a point on the x-axis due to a single wire is given by:
\[ B = \frac{\mu_0 I}{2\pi r} \]
where \( r \) is the distance from the wire to the point where we are calculating the magnetic field.

### Step 3: Calculate the Magnetic Field at a Point on the x-axis
Let’s consider a point \( P \) on the x-axis at a distance \( x \) from the origin. The distances from the two wires to point \( P \) will be:
- From the wire at \( y = A \): \( r_1 = \sqrt{x^2 + A^2} \)
- From the wire at \( y = -A \): \( r_2 = \sqrt{x^2 + A^2} \)

Since both currents are directed into the plane, the magnetic fields at point \( P \) due to both wires will be in opposite directions. Therefore, the total magnetic field \( B \) at point \( P \) is:
\[ B = B_1 - B_2 = \frac{\mu_0 I}{2\pi r_1} - \frac{\mu_0 I}{2\pi r_2} \]
Since \( r_1 = r_2 \), we can simplify this to:
\[ B = 0 \]
at the point directly on the x-axis when \( x = 0 \).

### Step 4: Analyze the Behavior of the Magnetic Field as \( x \) Changes
1. **At \( x = 0 \)**: The magnetic field \( B = 0 \).
2. **As \( x \to \infty \)**: The distance \( r \) becomes very large, and thus:
\[ B \to 0 \]
3. **At \( x = A \)**: We can substitute \( x = A \) into the magnetic field equation:
\[ B = \frac{\mu_0 I}{2\pi A} - \frac{\mu_0 I}{2\pi A} = 0 \]
4. **At \( x = -A \)**: Similarly, we find:
\[ B = 0 \]

### Step 5: Determine the Maximum Value of \( B \)
To find the maximum value of \( B \), we can differentiate the expression for \( B \) with respect to \( x \) and set it to zero. This will give us the points where \( B \) could be maximized.

### Step 6: Graph the Magnitude of \( B \)
The graph of the magnetic field \( B \) as a function of \( x \) will show:
- \( B = 0 \) at \( x = 0 \)
- \( B \) will reach a maximum value at \( x = A \) and \( x = -A \)
- \( B \to 0 \) as \( x \to \infty \) or \( x \to -\infty \)

### Conclusion
The graph will show a symmetrical behavior around the origin, with peaks at \( x = A \) and \( x = -A \), and it will touch the x-axis at \( x = 0 \) and as \( x \to \pm \infty \).

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