What is the magnetic field B at a distance r = 0.5 m from an infinitely long straight wire carrying a current

Question

What is the magnetic field B at a distance r = 0.5 m from an infinitely long straight wire carrying a current I = 10 A? (Given $\mu_0 = 4\pi 	imes 10^{-7}$ T m/A)

Examoogle · Accepted Answer

The correct answer is Option B: 4 	imes 10^{-6}	ext{ T}. The magnetic field at a distance from an infinitely long straight wire is calculated using the Biot-Savart Law derived formula. For a current of $I = 10	ext{ A}$ at $r = 0.5	ext{ m}$, the result is $B = 4 	imes 10^{-6}	ext{ T}$.

Step-by-Step Solution & Concepts:
The magnetic field at a distance from an infinitely long straight wire is calculated using the Biot-Savart Law derived formula. For a current of $I = 10	ext{ A}$ at $r = 0.5	ext{ m}$, the result is $B = 4 	imes 10^{-6}	ext{ T}$.

Working Principle: According to Ampere's Circuital Law, the magnetic field $B$ around a straight current-carrying wire is inversely proportional to the perpendicular distance from the wire. The formula used is $B = \frac{\mu_0 I}{2\pi r}$. By substituting the given values, $B = \frac{4\pi 	imes 10^{-7} 	imes 10}{2\pi 	imes 0.5} = \frac{4\pi 	imes 10^{-6}}{\pi} = 4 	imes 10^{-6}	ext{ T}$.

Key Formulas: $B = \frac{\mu_0 I}{2\pi r}$ — Magnetic field of an infinitely long straight wire; $\mu_0 = 4\pi 	imes 10^{-7} 	ext{ T m/A}$ — Permeability of free space

Key Points: The magnetic field lines are concentric circles centered on the wire.; The direction of the field is determined by the Right-Hand Thumb Rule.; Magnetic field strength decreases linearly with distance $r$.

Advantages: Provides a simple, predictable field configuration for calculations.

Disadvantages: Assumes an infinitely long wire, which is an idealization in practical laboratory settings.

Applications: Design of electromagnetic coils; Power transmission line analysis

Additional Info: The SI unit for magnetic flux density is the Tesla ($T$).. Option A results from miscalculating the $2\pi$ factor, while C and D are arithmetic errors.

Answer: B is correct — Using the formula $B = \frac{\mu_0 I}{2\pi r}$ with the provided values yields a magnetic field magnitude of $4 	imes 10^{-6}	ext{ T}$.

Exam Tip: Always ensure your distance $r$ is in meters and your current $I$ is in Amperes before plugging into the magnetic field formula to avoid unit conversion errors.

Examoogle · Answer

Option A: 2 	imes 10^{-6}	ext{ T}

Examoogle · Answer

Option C: 8 	imes 10^{-6}	ext{ T}

Examoogle · Answer

Option D: 1 	imes 10^{-6}	ext{ T}

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