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When a wire carries an electric current, it creates a magnetic field around it. This magnetic field is crucial in many applications, from powering our homes to driving electric motors. To calculate the magnetic field generated by a straight, current-carrying wire, we use the formula:

• \( B = \frac{\mu_0 I}{2\pi r} \)

In this equation:

• \( B \) represents the magnetic field in teslas (T).
• \( I \) is the current flowing through the wire in amperes (A).
• \( r \) is the distance from the wire where the magnetic field is being measured (in meters).
• \( \mu_0 \), the permeability of free space, is a constant value.

This formula allows us to understand how the magnetic field varies with distance from the wire. It's essential to remember that as you move further away from the wire, the magnetic field weakens. This inverse relationship is evident as \( r \) appears in the denominator of the formula.
Understanding this fundamental equation helps us predict how magnetic fields behave around current-carrying wires.

The permeability of free space, denoted as \( \mu_0 \), is a crucial constant in electromagnetism. It characterizes the magnetic influence of free space and appears in many fundamental equations. For current-carrying wires, it aids in calculating the produced magnetic field.
The value of \( \mu_0 \) is specifically defined as:

• \( \mu_0 = 4\pi \times 10^{-7} \ \mathrm{T} \cdot \mathrm{m/A} \)

This constant is the same for all spaces without any magnetic materials present. It essentially represents the ability of a vacuum to support magnetic fields. If you have a highly sensitive application or need a precise understanding of the wire's field, knowing \( \mu_0 \) is invaluable.
For practical uses, recognizing \( \mu_0 \) helps in calculating the influence of magnetic fields produced by electric currents in real-world scenarios. Always ensure the units are consistent, so calculations are accurate.

Calculating the distance from a wire where a specific magnetic field strength is detected involves rearranging the magnetic field formula for distance. This process helps in applications where knowing the precise position relative to a magnetic field source is vital.
From the formula \( B = \frac{\mu_0 I}{2\pi r} \), solving for the distance \( r \) involves:

• Rearranging to obtain \( r = \frac{\mu_0 I}{2\pi B} \).

Given values allow for practical calculations:

• Using \( \mu_0 = 4\pi \times 10^{-7} \ \mathrm{T} \cdot \mathrm{m/A} \)
• With a current \( I = 4.2 \ \mathrm{A} \)
• And a magnetic field \( B = 1.3 \times 10^{-5} \ \mathrm{T} \).

By plugging these into the formula, the calculated distance \( r \) is approximately 0.0646 meters.
This practical insight is beneficial for placing sensors, designing circuits, or troubleshooting electromagnetic interference. The ability to compute \( r \) provides a deeper understanding of the spatial dimensions of magnetic fields.