Question

Use Ampere’s law to derive the expression for the magnetic field due to a long straight current-carrying wire of infinite length.

Answer 1

Ampere's Law and Magnetic Field of a Current-Carrying Wire

Ampere's law states that the magnetic field \( \vec{B} \) generated by a current-carrying conductor can be determined using the equation:

\[ \oint_C \vec{B} \cdot d\vec{l} = \mu_0 I \]

Definitions:

• \( \oint_C \) denotes the line integral over a closed loop \( C \).
• \( \vec{B} \) is the magnetic field vector.
• \( d\vec{l} \) is an infinitesimal displacement vector along the loop.
• \( I \) is the total current enclosed by the loop.
• \( \mu_0 \) is the magnetic constant (permeability of free space).

Derivation of Magnetic Field for an Infinitely Long Straight Wire:

1. Assume the wire carries current \( I \).
2. The magnetic field is assumed to be circular and symmetric, meaning its magnitude \( B \) depends solely on the radial distance \( r \) from the wire.
3. A circular loop of radius \( r \), centered on the wire, is selected.

Due to symmetry, the magnetic field is tangential to the loop at all points, and its magnitude \( B \) is constant. Therefore, the line integral simplifies to:

\[ \oint_C \vec{B} \cdot d\vec{l} = B \oint_C dl = B (2 \pi r) \]

Applying Ampere’s law:

\[ B (2 \pi r) = \mu_0 I \]

Solving for \( B \):

\[ B = \frac{\mu_0 I}{2 \pi r} \]

Result: The magnetic field at a distance \( r \) from an infinitely long straight wire carrying a current \( I \) is:

\[ B = \frac{\mu_0 I}{2 \pi r} \]