Magnetic Field Due to a Straight Conductor Carrying Current

1. Concept Overview

A straight wire carrying electric current always produces a magnetic field around it.
This field forms concentric circular loops centered on the wire.

The magnetic field at a distance r from a long, straight conductor carrying current I is:

[B] [= \left( \dfrac{\mu_{0} I}{2\pi r} \right)]

Intuitive Understanding (Beginner Friendly)

• Current is simply moving charges.
• Moving charges create a magnetic influence around them.
• For a straight wire, this influence spreads out in the form of circles around the conductor.
• The closer you are to the wire, the stronger the magnetic field.
• If the current is increased, the circles become “stronger.”
• If current direction is reversed, magnetic field direction reverses.

This experiment was originally observed by Oersted, forming the foundation for the modern study of electromagnetism.

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2. Clear Explanation and Mathematical Derivation

We use the Biot–Savart Law, which gives magnetic field contribution from a tiny current element:

[dB] [= \left( \dfrac{\mu_{0}}{4\pi} \dfrac{I dl \sin\theta}{r^{2}} \right)]

For a very long straight wire:

• The magnetic field at point P makes [\theta = 90^\circ].
• So [\sin\theta = 1].

Let distance from point P to wire be [r].

For a long wire:

[B = \displaystyle \int dB] [= \dfrac{\mu_{0} I}{4\pi r} \displaystyle \int_{-\infty}^{\infty} \dfrac{dl}{r}]

Using geometry:

[\displaystyle \int dB] [= \dfrac{\mu_{0} I}{4\pi r} (2)]

Thus:

[B] [= \left( \dfrac{\mu_{0} I}{2\pi r} \right)]

Direction:

Use Right-Hand Thumb Rule:

• Thumb → direction of current
• Fingers curl → direction of magnetic field lines (circular)

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3. Dimensions and Units

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4. Key Features

• Magnetic field decreases as 1/r from the wire.
• Field strength is directly proportional to current.
• Field lines are concentric circles around the conductor.
• Direction follows Right-Hand Thumb Rule.
• For multiple wires, fields add or subtract using vector addition.
• Practical applications:
  • Electromagnets
  • Electric motors
  • Induction furnaces
  • MRI coils

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5. Important Formulas to Remember

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6. Conceptual Questions with Solutions

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7. FAQ / Common Misconceptions

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8. Practice Questions (with Step-by-Step Solutions)

Q1.

Find magnetic field at a distance 5 cm from a wire carrying 10 A.

[
B = \left( \dfrac{\mu_{0} I}{2\pi r} \right)
]

[B = \left( \dfrac{4\pi\times10^{-7} \times 10}{2\pi \times 0.05} \right)]

[
B = 4 \times 10^{-5} \text{T}
]

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Q2.

What happens to B if distance increases from 2 cm to 6 cm?

[
B \propto \dfrac{1}{r}
]

Field becomes one-third.

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Q3.

A point is 0.2 m from a wire carrying 5 A. Find B.

[B] [= \left( \dfrac{4\pi \times 10^{-7} \times 5}{2\pi \times 0.2} \right)]

[
B = 5 \times 10^{-6} \text{T}
]

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Q4.

Two parallel wires carry currents in same direction. What happens to magnetic fields between them?

Fields cancel in the region between wires.

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Q5.

At what distance from a 8 A wire will B = [1 \times 10^{-5}] T?

[r] [= \left( \dfrac{\mu_{0} I}{2\pi B} \right)]

[r] [= \left( \dfrac{4\pi \times 10^{-7} \times 8}{2\pi \times 10^{-5}} \right)]

[
r = 0.016\ \text{m}
]