Question

A straight wire of diameter $0.4 \text{ mm}$ carrying a current of $2 \text{ A}$ is replaced by another wire of $0.8 \text{ mm}$ diameter carrying the same current. The magnetic field at distance (R) from both the wires is '$B_1$' and '$B_2$' respectively. The relation between $B_1$ and $B_2$ is

• A. $B_1 = \frac{B_2}{2}$
• B. $B_1 = B_2$
• C. $B_1 = 2 B_2$
• D. $B_1 = \frac{B_2}{3}$

Hint:

Don't fall into the trap of using wire dimensions for external fields! A wire's radius or diameter changes the internal magnetic field gradient ($\text{inside } B \propto r$), but for any point outside the wire, the system functions exactly like an infinitely thin line current.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question compares the magnetic fields $B_1$ and $B_2$ produced at an external point located at a fixed distance $R$ away from the central axis of two different straight wires. The second wire has double the diameter of the first, but carries the exact same current.

Step 2: Key Formula or Approach:
According to Ampere's Law, the magnetic field $B$ at an external point located at a perpendicular distance $R$ from a long, straight current-carrying conductor is given by:
$$B = \frac{\mu_0 I}{2\pi R}$$

Step 3: Detailed Explanation:
Looking closely at the Ampere's Law formula, the magnetic field strength at any external point is completely determined by two factors:
1. The magnitude of the electrical current flowing through the conductor ($I$).
2. The radial distance of the observation point measured from the wire's center ($R$).
The physical thickness or diameter of the conducting wire has zero effect on the external field expression, provided the point lies entirely outside the wire's cross-section.
Since the problem specifies that both wires carry the exact same current ($I_1 = I_2 = 2 \text{ A}$) and the magnetic fields are measured at the exact same distance $R$, the field values must be identical:
$$B_1 = B_2$$

Step 4: Final Answer:
The relation between the magnetic fields is $B_1 = B_2$, which corresponds to option (B).