The magnetic field at a point d distance away from long wire due to the electric current I in it is __________

Question

The magnetic field at a point d distance away from long wire due to the electric current I in it is ____________

Examoogle · Accepted Answer

Option C: $\frac{\mu_0 I}{2\pi r}$

Explanation & Concepts:
The magnetic field intensity at a distance $r$ from an infinitely long, straight conductor carrying a steady current $I$ is inversely proportional to the distance and directly proportional to the current. According to Ampere's Circuital Law, this relationship is given by $B = \frac{\mu_0 I}{2\pi r}$. Thus, Option C represents the correct mathematical expression for the magnetic field.

Working Principle: When an electric current flows through a straight conductor, it produces magnetic flux lines in concentric circles around the wire. Applying Ampere's Circuital Law around a closed circular path of radius $r$ gives $\oint B \cdot dl = B(2\pi r) = \mu_0 I_{	ext{enc}}$, which yields $B = \frac{\mu_0 I}{2\pi r}$.

Key Formulas: $B = \frac{\mu_0 I}{2\pi r}$ — Magnetic field at distance $r$ from a long straight current-carrying wire; $\oint B \cdot dl = \mu_0 I_{	ext{enc}}$ — Ampere's Circuital Law

Key Points: The magnetic field lines form concentric circles around the conductor.; The magnitude of the magnetic field $B$ decreases linearly as the distance $r$ increases ($B \propto 1/r$).; The direction of the magnetic field is determined by the Right-Hand Thumb Rule.

Applications: Calculating magnetic forces between parallel current-carrying conductors in transmission lines.; Designing inductor coils and understanding electromagnetic interference (EMI) around heavy current conductors.

Additional Info: Option A ($\frac{\mu_0 I}{2r}$) represents the magnetic field at the center of a circular current-carrying loop of radius $r$.. Option C ($\frac{\mu_0 I}{2\pi r}$) is correct for a long straight conductor at distance $r$ (or $d$).

Answer: C is correct — The magnetic field due to a long current-carrying conductor at a distance $r$ is given by $B = \frac{\mu_0 I}{2\pi r}$.

Exam Tip: Always distinguish between a straight conductor ($B = \frac{\mu_0 I}{2\pi r}$) and a circular loop center ($B = \frac{\mu_0 I}{2r}$); the factor of $\pi$ appears in the denominator for straight wires due to the circular perimeter $2\pi r$ integration path.

Examoogle · Answer

Option A: $\frac{\mu_0 I}{2r}$

Examoogle · Answer

Option B: $\frac{\mu_0 I}{r}$

Examoogle · Answer

Option D: $\frac{\mu_0 I}{\pi r}$

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