A circular loop of radius r carries a current i. How should a long, straight wire carrying a current 4i be placed in the plane of the circle so that the magnetic field at the center becomes zero?

Answer

To solve the problem of how to place a long straight wire carrying a current of \(4i\) in the plane of a circular loop of radius \(r\) (which carries a current \(i\)) such that the magnetic field at the center of the loop becomes zero, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Magnetic Fields**:
- The magnetic field at the center of a circular loop carrying a current \(i\) is given by the formula:
\[ B_{\text{loop}} = \frac{\mu_0 i}{2r} \]
- The magnetic field due to a long straight wire carrying a current \(I\) at a distance \(d\) from the wire is given by:
\[ B_{\text{wire}} = \frac{\mu_0 I}{2\pi d} \]

2. **Set Up the Equation for Zero Magnetic Field**:
- For the magnetic field at the center of the loop to be zero, the magnetic field produced by the circular loop must be equal in magnitude and opposite in direction to the magnetic field produced by the straight wire:
\[ B_{\text{loop}} + B_{\text{wire}} = 0 \]
- This can be rearranged to:
\[ B_{\text{loop}} = -B_{\text{wire}} \]

3. **Substitute the Magnetic Field Formulas**:
- Substituting the expressions for \(B_{\text{loop}}\) and \(B_{\text{wire}}\):
\[ \frac{\mu_0 i}{2r} = \frac{\mu_0 (4i)}{2\pi d} \]

4. **Cancel Common Terms**:
- We can cancel \(\mu_0\) and \(i\) from both sides (assuming \(i \neq 0\)):
\[ \frac{1}{2r} = \frac{4}{2\pi d} \]

5. **Simplify the Equation**:
- Simplifying the equation gives:
\[ \frac{1}{2r} = \frac{2}{\pi d} \]

6. **Cross-Multiply to Solve for \(d\)**:
- Cross-multiplying gives:
\[ \pi d = 4r \]
- Therefore, solving for \(d\):
\[ d = \frac{4r}{\pi} \]

7. **Conclusion**:
- The distance \(d\) from the center of the circular loop to the straight wire should be \(\frac{4r}{\pi}\) for the magnetic field at the center of the loop to be zero.

### Final Answer:
The long straight wire carrying a current \(4i\) should be placed at a distance of \(\frac{4r}{\pi}\) from the center of the circular loop.