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 Due to a Current-Carrying Wire

According to Ampere’s law, the magnetic field \( \vec{B} \) due to a current-carrying conductor can be derived using the following equation:

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

Where:

• \( \oint_C \) represents the line integral around a closed loop \( C \)
• \( \vec{B} \) is the magnetic field at a point
• \( d\vec{l} \) is an infinitesimal vector along the loop
• \( I \) is the total current passing through the loop
• \( \mu_0 \) is the permeability of free space

Assumptions for Deriving the Magnetic Field Due to an Infinitely Long Straight Wire:

1. The wire carries a current \( I \).
2. The magnetic field around the wire is circular and symmetric, so its magnitude \( B \) depends only on the radial distance \( r \) from the wire.
3. A circular loop of radius \( r \) is chosen, centered at the wire.

By symmetry, the magnetic field at every point on the loop is tangent to the circle, and the magnitude of \( \vec{B} \) is constant at all points on the loop. Hence, the line integral becomes:

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

Using Ampere’s law:

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

Solving for \( B \):

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

Final Answer: Thus, the magnetic field at a distance \( r \) from an infinitely long straight wire carrying a current \( I \) is:

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