Problem 46 - College Physics

Question

\(\bullet\) If the magnetic field due to a long, straight current-carrying wire has a magnitude \(B\) at a distance \(R\) from the wire's center, how far away must you be (in terms of \(R\) ) for the magnetic field to decrease to \(B / 3 ?\)

Step-by-Step Solution

Answer

You must be at a distance of \( 3R \) from the wire for the magnetic field to decrease to \( B/3 \).

Step 1: Understanding the Relationship

The magnetic field due to a long, straight current-carrying wire is given by the formula \( B = \frac{\mu_0 I}{2 \pi r} \), where \( B \) is the magnetic field, \( \mu_0 \) is the permeability of free space, \( I \) is the current, and \( r \) is the distance from the wire. This formula shows that the magnetic field is inversely proportional to the distance \( r \).

Step 2: Setting the Equation for a Reduced Field

For the field to decrease to \( \frac{B}{3} \), set up the equation for the new magnetic field as \( \frac{\mu_0 I}{2 \pi r'} = \frac{1}{3} \times \frac{\mu_0 I}{2 \pi R} \) where \( r' \) is the new distance we need to find.

Step 3: Solving for the New Distance

By simplifying the equation, \( \frac{1}{r'} = \frac{1}{3} \times \frac{1}{R} \), hence multiplying both sides by 3 and \( r' \), we get \( r' = 3R \).

Key Concepts

Inverse Proportionality

When it comes to understanding the behavior of magnetic fields created by current-carrying wires, inverse proportionality plays a critical role. This means that if one value increases, the other decreases. For magnetic fields, this is all about the relationship between the magnetic field strength and the distance from the wire. The formula to determine the magnetic field around a long, straight wire is \[ B = \frac{\mu_0 I}{2 \pi r} \]
where:

• \( B \) is the magnetic field strength
• \( \mu_0 \) is the permeability of free space, a constant
• \( I \) represents the current flowing through the wire
• \( r \) is the distance from the wire

This formula indicates that as the distance \( r \) increases, the magnetic field \( B \) decreases, since they are inversely proportional. So, doubling the distance from the wire will halve the magnetic field strength. This is a fundamental concept in the study of magnetism and electromagnetism.

Distance and Magnetic Field Relationship

The relationship between distance and magnetic field strength is a pivotal aspect of understanding electromagnetism. As already established through inverse proportionality, the magnetic field weakens as the distance from the wire increases.
For instance, suppose you are standing at a distance \( R \) from a wire, experiencing a magnetic field \( B \). If you want the magnetic field to be only one-third as strong, you need to increase your distance.
We can find this new distance \( r' \) by setting the equation for a weaker magnetic field, \[ \frac{\mu_0 I}{2 \pi r'} = \frac{1}{3} \times \frac{\mu_0 I}{2 \pi R} \]Simplifying, we find that \[ \frac{1}{r'} = \frac{1}{3} \times \frac{1}{R} \]From this, it turns out that \[ r' = 3R \]So, to make the field decrease to one-third its original strength, you should be three times farther from the wire than your original distance.

Magnetic Field Strength

Magnetic field strength is an essential factor in electromagnetism and affects many applications in physics and engineering. It describes the magnitude of the magnetic field produced by a current-carrying conductor. A stronger current or closer proximity to the wire results in a stronger magnetic field.
To impact field strength intentionally, like reducing it to \( \frac{B}{3} \), you must increase your distance as detailed earlier. This principle is crucial in designing magnetic devices and systems, as understanding how the magnetic field behaves helps in controlling and optimizing performance.
Moreover, the concept is significant in devising safety measures in environments where magnetic fields are in use, ensuring that the exposure is within safe and acceptable limits due to the inverse proportionality relationship."}]}]}