Question

What is Ampere's circuital law? Using it, obtain the formula for the magnetic field produced due to a straight current-carrying conductor of infinite length.

Hint:

Ampere's circuital law is useful for calculating the magnetic field around symmetrical current distributions. For a straight conductor, the field decreases with the distance from the conductor.

Answer 1

Ampere's Circuital Law:
Ampere's circuital law states that the line integral of the magnetic field \( \mathbf{B} \) around a closed loop is proportional to the total current \( I \) passing through the loop. Mathematically, it is given by:
\[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enc}}, \] where:
- \( \oint \mathbf{B} \cdot d\mathbf{l} \) is the line integral of the magnetic field around a closed loop,
- \( \mu_0 \) is the permeability of free space (\( \mu_0 = 4 \pi \times 10^{-7} \, \text{T} \cdot \text{m}/\text{A} \)),
- \( I_{\text{enc}} \) is the enclosed current by the loop.
Magnetic Field Due to a Straight Current-Carrying Conductor:
Consider a long, straight conductor carrying a current \( I \). To find the magnetic field at a distance \( r \) from the conductor, we apply Ampere's circuital law. We take a circular loop of radius \( r \) around the conductor, as the magnetic field produced by a straight conductor is symmetric in nature and forms concentric circles around the conductor.
The magnitude of the magnetic field at a distance \( r \) from the wire is constant along the circular loop, and the direction of the magnetic field is tangential to the loop. Therefore, the line integral simplifies to:
\[ \oint \mathbf{B} \cdot d\mathbf{l} = B \cdot 2\pi r, \] where \( B \) is the magnetic field at a distance \( r \) from the wire. Using Ampere's law:
\[ B \cdot 2\pi r = \mu_0 I. \] Solving for \( B \), we get:
\[ B = \frac{\mu_0 I}{2 \pi r}. \] Thus, the magnetic field at a distance \( r \) from a long, straight current-carrying conductor is:
\[ B = \frac{\mu_0 I}{2 \pi r}. \]