Understanding the Magnetic Field of a Straight Current-Carrying Wire

The question asks about how the magnetic field produced by a current-carrying straight wire changes as you move away from it. This is a fundamental concept in electromagnetism, specifically described by laws like the Biot-Savart Law or Ampere's Law.

For a long, straight wire carrying a steady electric current (let's call the current \(I\)), the magnetic field (\(B\)) at a point located a distance \(r\) away from the wire is given by a specific formula. Assuming the wire is in a vacuum or air, the formula is:

\( B = \frac{\mu_0 I}{2 \pi r} \)

Let's break down this formula:

• \(B\) represents the strength of the magnetic field at the point.
• \( \mu_0 \) is the permeability of free space, a constant value.
• \(I\) is the amount of current flowing through the wire.
• \(r\) is the perpendicular distance from the wire to the point where the magnetic field is being measured.
• \(2\pi\) is a constant.

Looking at the formula \( B = \frac{\mu_0 I}{2 \pi r} \), we can see how the magnetic field \(B\) depends on the distance \(r\). The term \(\frac{\mu_0 I}{2 \pi}\) is constant for a given wire carrying a steady current \(I\). Therefore, the formula shows that \(B\) is proportional to \( \frac{1}{r} \).

\( B \propto \frac{1}{r} \)

This relationship, \( B \propto \frac{1}{r} \), means that the magnetic field strength is inversely proportional to the distance \(r\) from the wire. In simpler terms:

• If you double the distance (\(r\)) from the wire, the magnetic field strength (\(B\)) becomes half of what it was.
• If you halve the distance (\(r\)) from the wire, the magnetic field strength (\(B\)) becomes double what it was.

The magnetic field gets weaker as you move away from the wire, and stronger as you move closer to the wire. This inverse dependence on distance is a key characteristic of the magnetic field around a straight current-carrying wire.

Now let's look at the options in light of this understanding:

• Option 1: says the magnetic field depends "inversely on the distance from it". This matches our finding \( B \propto \frac{1}{r} \).
• Option 2: says the magnetic field depends "directly on the distance from it". This would mean \( B \propto r \), which is not supported by the formula.
• Option 3: says the dependence is "inversely at short distances and directly at large distances from it". The formula \( B = \frac{\mu_0 I}{2 \pi r} \) shows a consistent inverse dependence \( B \propto \frac{1}{r} \) regardless of whether the distance \(r\) is short or large (as long as the point is outside the wire).
• Option 4: says the dependence is "directly on the distance (at short distances) and inversely on the distance (at long distances) from it". Similar to option 3, the formula indicates a purely inverse relationship outside the wire, not a mixed one depending on distance magnitude.

Based on the formula derived from electromagnetic principles, the magnetic field produced by a current-carrying straight wire at a point outside the wire is indeed inversely proportional to the distance from the wire.

Revision Table: Magnetic Field Dependence

Additional Information: Magnetic Field of Wires

Beyond the magnitude, the magnetic field produced by a straight current-carrying wire also has a direction. This direction can be found using the Right-Hand Rule. If you point the thumb of your right hand in the direction of the current flow in the wire, your fingers will curl around the wire in the direction of the magnetic field lines. The magnetic field lines form concentric circles around the wire.

This inverse dependence on distance is characteristic of magnetic fields produced by long, straight current sources. For other shapes of current loops or wires, the dependence on distance can be different and more complex. For example, near the center of a current loop, the field is relatively uniform, but far away, it might decrease faster with distance (like \(1/r^3\)). However, for a straight wire, it's consistently \(1/r\) outside the wire.