To find the magnetic field at the origin due to the infinite collection of current-carrying conductors, we can follow these steps:
### Step 1: Identify the configuration of the conductors
We have two sets of conductors:
1. The first set carries a current \( I \) outward (positive direction) and is located at \( x = a, 3a, 5a, \ldots \).
2. The second set carries a current \( I \) inward (negative direction) and is located at \( x = 2a, 4a, 6a, \ldots \).
### Step 2: Determine the magnetic field due to the first set of conductors
The magnetic field \( B \) at a distance \( r \) from a long straight conductor carrying current \( I \) is given by:
\[
B = \frac{\mu_0 I}{2 \pi r}
\]
For the first set of conductors, the distances from the origin are \( a, 3a, 5a, \ldots \). The magnetic field contributions from these conductors at the origin can be expressed as:
\[
B_{\text{outward}} = \frac{\mu_0 I}{2 \pi a} + \frac{\mu_0 I}{2 \pi (3a)} + \frac{\mu_0 I}{2 \pi (5a)} + \ldots
\]
This can be simplified to:
\[
B_{\text{outward}} = \frac{\mu_0 I}{2 \pi a} \left(1 + \frac{1}{3} + \frac{1}{5} + \ldots\right)
\]
### Step 3: Determine the magnetic field due to the second set of conductors
For the second set of conductors, the distances from the origin are \( 2a, 4a, 6a, \ldots \). The magnetic field contributions from these conductors at the origin can be expressed as:
\[
B_{\text{inward}} = -\left(\frac{\mu_0 I}{2 \pi (2a)} + \frac{\mu_0 I}{2 \pi (4a)} + \frac{\mu_0 I}{2 \pi (6a)} + \ldots\right)
\]
This can be simplified to:
\[
B_{\text{inward}} = -\frac{\mu_0 I}{2 \pi (2a)} \left(1 + \frac{1}{2} + \frac{1}{3} + \ldots\right)
\]
### Step 4: Combine the contributions from both sets
Now, we need to combine the contributions from both sets:
\[
B_{\text{net}} = B_{\text{outward}} + B_{\text{inward}}
\]
Substituting the expressions we derived:
\[
B_{\text{net}} = \frac{\mu_0 I}{2 \pi a} \left(1 + \frac{1}{3} + \frac{1}{5} + \ldots\right) - \frac{\mu_0 I}{2 \pi (2a)} \left(1 + \frac{1}{2} + \frac{1}{3} + \ldots\right)
\]
### Step 5: Recognize the series
The series \( 1 + \frac{1}{3} + \frac{1}{5} + \ldots \) is the sum of the reciprocals of the odd integers, while \( 1 + \frac{1}{2} + \frac{1}{3} + \ldots \) is the harmonic series. The difference of these series can be expressed in terms of logarithmic functions.
### Step 6: Final expression for the magnetic field
Using the properties of logarithmic expansions, we can express the net magnetic field at the origin as:
\[
B_{\text{net}} = \frac{\mu_0 I}{2 \pi a} \ln(2)
\]
Thus, the final answer is:
\[
B_{\text{net}} = \frac{\mu_0 I}{2 \pi a} \ln(2)
\]
