Magnetic Field of an Infinite Wire

Magnetic Field Strength (H) of an Infinite Wire

Deriving the magnetic field produced by a current-carrying conductor using the Biot-Savart Law.

Vector Geometry Diagram

The magnetic field H always forms a circle (a_φ) perpendicular to the current flow.

1. The Fundamental Law

To find the magnetic field intensity (H) at a point P near a wire, we start with the Biot-Savart Law for a differential current element:

dH = (I dL × aᵣ) / (4πr²)

Defining the Parameters:

• dL: The small segment of the wire, defined as dz a_z.
• r: The distance from the segment to point P, which forms the hypotenuse of a triangle: √(ρ² + z²).
• aᵣ: The unit vector pointing from the wire to the observation point.

2. Solving the Vector Cross Product

The direction of the magnetic field is determined by the cross product of the current direction and the distance vector. In cylindrical coordinates:

Cross Product Logic:
(a_z) × (a_ρ) = a_φ (Direction of the field)
(a_z) × (a_z) = 0 (Parallel components produce no field)

This tells us that the magnetic field curls around the wire in the azimuthal direction (a_φ).

3. The Integration Process

To find the total field from an infinite wire, we integrate from -∞ to +∞. The notes use a trigonometric substitution to simplify the math:

Let z = ρ tan(θ)
dz = ρ sec²(θ) dθ

By substituting these values, the complex integral converts into a simple cosine integration over the angles -π/2 to π/2:

H = (I / 4πρ) ∫ cos(θ) dθ = (I / 4πρ) [sin(θ)]

4. Final Formulas

For an Infinite Wire:

After evaluating the limits, we arrive at the standard formula for an infinitely long conductor:

H = (I / 2πρ) a_φ

For a Finite Wire Extension

If the wire is not infinite, we use the angles (α₁, α₂) from the observation point to the ends of the wire:

H = [I / 4πρ] (sin α₂ + sin α₁) a_φ

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Note: This result is consistent with Ampere's Circuital Law, which states that the line integral of H around a closed path is equal to the current enclosed (∮ H · dL = I).

From Finite Wire to Ampere’s Law

You asked if the finite wire formula can be formulated into ∮ H · dL = I. The answer is yes, but it requires the wire to become "infinite." Here is the logical bridge:

Step 1: The Infinite Limit

In the formula for a finite wire length:

H = [I / 4πρ] (sin α₂ + sin α₁) a_φ

As the wire becomes infinitely long, the angles α₁ and α₂ both approach 90° (π/2). Since sin(90°) = 1, the term becomes (1 + 1) = 2.

Infinite Result:
H = [I / 4πρ] * 2 = I / 2πρ

Step 2: Connecting to Ampere's Law

Now, let's test if this result satisfies Ampere’s Circuital Law. We take a circular path of radius ρ around the wire. Along this path, the differential length is dL = ρ dφ a_φ.

∮ H · dL = ∫₀²π (I / 2πρ a_φ) · (ρ dφ a_φ)

Notice what happens during the dot product calculation:

• The unit vectors a_φ · a_φ = 1.
• The distance ρ in the numerator and denominator cancel out.
• The constants move outside the integral.

∮ H · dL = (I / 2π) ∫₀²π dφ = (I / 2π) * 2π = I

The Conclusion:
The derivation using the Biot-Savart Law for a finite wire is perfectly consistent with Ampere's Law. When the geometry is infinite, the complex trigonometric term simplifies exactly into the reciprocal of the circumference (2πρ), proving that the total enclosed current is indeed I.

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Practical Tip: Use Biot-Savart for complex segments or finite wires, and use Ampere's Law for infinite, symmetric systems to save time!

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