Magnetic field due to a straight wire

The magnetic field generated by a straight wire carrying an electric current is a fundamental concept in physics, particularly in electromagnetism. It’s quantified by Ampère’s law, a cornerstone of Maxwell’s equations, which govern all electromagnetic phenomena.

Magnetic Field Due to a Straight Wire

The magnetic field generated by a straight wire carrying an electric current is a fundamental concept in physics, particularly in electromagnetism. It’s quantified by Ampère’s law, a cornerstone of Maxwell’s equations, which govern all electromagnetic phenomena.

Ampère’s Law and the Biot-Savart Law

Ampère’s law, combined with the Biot-Savart law, provides an equation to calculate the magnetic field (B) at a point due to a current (I) flowing through a straight wire.

1. The Biot-Savart Law is an equation that describes the magnetic field created by a current-carrying wire:

B = μ₀/4π ∫ Idl x ⃗r / r³

2. Here,

• μ₀ is the permeability of free space,

• dl is a small element of the wire carrying the current,

• ⃗r is the position vector from the element dl to the point in space where the magnetic field is being calculated,

• r is the distance between dl and the point.

3. The cross product (dl x ⃗r) indicates the direction of the magnetic field, which is always perpendicular to the plane formed by dl and ⃗r.

Magnetic Field Due to a Straight Wire

For a long, straight wire carrying current I, the magnetic field at a distance r from the wire is given by:

B = μ₀I / 2πr

1. This equation is a simplified form of the Biot-Savart law, derived by considering the wire to be composed of many small current elements.

• The direction of the magnetic field B is given by the right-hand rule. If the thumb of the right hand points in the direction of the current, the fingers curl in the direction of B.

Thus, a straight wire carrying a current produces a magnetic field that wraps around the wire in concentric circles, with its strength decreasing with increasing distance from the wire.

Example Calculation

Let’s consider a practical example to understand the calculation of the magnetic field due to a straight wire.

Suppose a wire is carrying a current of I = 10 Amps, and we want to find the magnetic field B at a distance r = 0.05 meters (or 5 cm) from the wire.

1. First, we note the value of the permeability of free space (μ₀), which is a constant equal to 4π x 10^-7 T m/A.

2. Then, we use the equation for the magnetic field due to a long, straight wire, which is:

B = μ₀I / 2πr

3. Substituting the given values into the equation, we get:

B = (4π x 10^-7 T m/A * 10 A) / (2π * 0.05 m)

4. Calculating the above expression, we find the magnetic field B at the specified location to be approximately 4 x 10^-5 Tesla.

Thus, this example demonstrates how to calculate the magnetic field produced by a straight wire carrying a current, using the derived equation from Ampère’s law and the Biot-Savart law.