Question

The maximum magnetic field produced by a current of 12 A passing through a copper wire of diameter 1.2 mm is

• A. 2 mT
• B. 4 mT
• C. 1.5 mT
• D. 8 mT

Hint:

The magnetic field produced by a current-carrying wire is strongest at the surface of the wire and decreases inversely with distance ($1/r$) outside the wire, making it a key concept in electromagnetism.

Answer 1

The Correct Option is B

Let’s break this down step by step to calculate the maximum magnetic field produced by the current in the wire and determine why option (2) is the correct answer.
Step 1: Understand the formula for the magnetic field produced by a current-carrying wire
The magnetic field $B$ at a distance $r$ from a long, straight wire carrying current $I$ is given by Ampere’s Law in the form:
\[ B = \frac{\mu_0 I}{2 \pi r} \]
where:

• $\mu_0$ is the permeability of free space, $\mu_0 = 4\pi \times 10^{-7} \, \text{T m A}^{-1}$,
• $I$ is the current,
• $r$ is the distance from the center of the wire.

The term "maximum magnetic field" typically means the field at the surface of the wire (i.e., at $r = \text{radius of the wire}$).
Step 2: Identify the given values and calculate the radius

• Current, $I = 12 \, \text{A}$
• Diameter of the wire, $d = 1.2 \, \text{mm} = 1.2 \times 10^{-3} \, \text{m}$
• Radius of the wire, $r = \frac{d}{2} = \frac{1.2 \times 10^{-3}}{2} = 0.6 \times 10^{-3} \, \text{m} = 0.0006 \, \text{m}$
• $\mu_0 = 4\pi \times 10^{-7} \, \text{T m A}^{-1}$

Step 3: Calculate the maximum magnetic field at the surface of the wire
Substitute the values into the formula:
\[ B = \frac{\mu_0 I}{2 \pi r} \]
\[ B = \frac{(4\pi \times 10^{-7}) \times 12}{2 \pi \times 0.0006} \]
Simplify:
\[ B = \frac{4 \times 10^{-7} \times 12}{0.0006 \times 2} \]
\[ B = \frac{4 \times 12 \times 10^{-7}}{0.0012} \]
\[ B = \frac{48 \times 10^{-7}}{0.0012} \]
\[ B = \frac{48}{1.2} \times 10^{-4} \]
\[ B = 40 \times 10^{-4} \, \text{T} \]
\[ B = 4 \times 10^{-3} \, \text{T} \]
\[ B = 4 \, \text{mT} \quad (\text{since } 1 \, \text{mT} = 10^{-3} \, \text{T}) \]
Step 4: Confirm the correct answer
The calculated magnetic field at the surface of the wire is 4 mT, which matches option (2). This confirms that the "maximum magnetic field" refers to the field at the wire’s surface, as expected for this type of problem.
Thus, the correct answer is (2) 4 mT.