Derivation and Formula for Magnetic Field Around a Straight Wire
The magnetic field due to straight wire is a foundational concept in electromagnetism, playing a key role in how current influences its environment. Imagine overhead power lines or phone chargers—each wire generates a magnetic field around itself when carrying electric current, an idea first proven by Oersted’s experiment. This principle forms a base for more complex analyses in JEE Main Physics.
Magnetic Field Due to Straight Wire: Principle and Law
Whenever current flows through a straight conductor, it produces a magnetic field forming concentric circles around the wire. The strength of this field is highest close to the wire and drops as you move away. This effect is governed by the Biot-Savart law and Ampere’s circuital law. Using the right-hand thumb rule, you determine the field’s direction: point your thumb in the direction of conventional current, and your fingers curl along the field lines.
This concept extends from class 10 basics to advanced JEE numericals. For field calculation near a wire of any length, formulae can be adapted. For infinite wires, the magnetic field is uniform along circles centered on the wire. For finite and semi-infinite wires, angles and geometry matter more.
- Current carrying wire magnetic field follows right-hand thumb rule.
- The field direction flips if current reverses.
- Applications include solenoids, sensors, and many JEE questions.
- Vedantu notes help clarify these derivations and formulae for exams.
Magnetic Field Due to Straight Wire: Formulae and Derivation
| Type of Wire | Magnetic Field Formula | Symbols |
|---|---|---|
| Infinite straight wire | B = (μ₀I)/(2πr) | B: field, μ₀: permeability, I: current, r: distance |
| Finite straight wire | B = (μ₀I)/(4πr) [sinθ₁ + sinθ₂] | θ₁, θ₂: angles at ends (see diagram) |
| Semi-infinite wire | B = (μ₀I)/(4πr) | One end at point, other extends to infinity |
Derivation (Infinite Wire): Applying the Biot-Savart law for a wire segment and integrating yields B = (μ₀I)/(2πr) for a point at distance r from a very long wire. Key assumptions: thin wire, steady current, and point outside the wire.
For a cylinder or wire of infinite length, use this formula directly in most JEE problems. For a finite length, you must include the angle terms, matching the diagram or question specifics.
Example: If 10 A passes through an infinitely long wire, and you measure at 5 cm (0.05 m), the field is B = (4π × 10-7 × 10)/(2π × 0.05) = 2 × 10-5 T.
Direction and Pattern: Right-Hand Thumb Rule and Field Lines
The direction of the magnetic field due to straight wire always forms circles centered on the wire. Use the right-hand thumb rule: thumb points with the current, fingers curl in field direction. Reversing current flips the field pattern; this is common in many JEE circuit setups.
The pattern is seen in iron filings experiments or via compass needles, which arrange themselves tangentially to the field lines. Closer to the wire, field lines are denser, indicating stronger field.
- Each point around the wire at fixed distance r experiences the same field magnitude.
- Field direction is tangent to the circular line at your observation point.
- If you use a right hand rule guide from Vedantu, you get clear visual support.
- Field reverses for opposite current flow—watch sign conventions in exams.
Experiment, Applications, and Common Pitfalls in JEE
The classic observation is the Oersted experiment: place a straight current-carrying conductor over a magnetic compass. When current flows, the compass needle turns, proving a magnetic field forms perpendicular to current direction. This core effect is asked directly and indirectly in JEE experimental skills sections.
To check yourself, try these steps with a wire, power supply, and a few compass needles spaced at different r values. Note how field strength decreases as you move further from the wire.
- Connect the straight wire to a low-voltage source.
- Place compass needles near the wire at different radial distances.
- Switch on current; observe needle alignment forming circles around the wire.
- Reverse current; note reversal in needle direction.
This experimental setup is crucial for quick Board and JEE revision.
- Common errors include omitting the 2πr factor, especially in infinite wire calculations.
- Use Biot-Savart law for complex geometries or short wire segments.
- For practice, go through topical mock tests to see field direction and formula applications.
- To compare with other shapes, see magnetic field due to a circular loop.
- For advanced properties, explore magnetic moment and related pages.
| Quick Reference | Key Details |
|---|---|
| Infinite straight wire field | B = (μ₀I)/(2πr); circular field lines; right-hand rule |
| Finite/semi-infinite wire field | Add angle terms: B = (μ₀I)/(4πr)[sinθ₁ + sinθ₂] |
| Field strength falls with r | Halving r doubles B; always check units |
| Field direction | Right-hand thumb: thumb = current, fingers = field |
| JEE tips | Mind sign conventions, angles, and geometry in numericals |
Understanding the magnetic field due to straight wire unlocks many advanced topics, such as solenoid fields, electromagnetic induction, and charged particle motion in fields. Avoid shortcuts—check sign, geometry, and units in every problem.
For full conceptual clarity, revisit current electricity basics and field laws. Combine this knowledge with circular loop and solenoid field patterns from toroid or solenoid field articles for a strong JEE Main foundation.
Vedantu Physics experts continually update topic pages to align with the latest JEE Main syllabus, providing trustworthy strategies, stepwise derivations, and plenty of practice links for full exam confidence.
FAQs on Magnetic Field Due to a Straight Current-Carrying Wire
1. What is the formula for the magnetic field due to a straight wire?
The magnetic field (B) at a distance r from a long straight wire carrying current I is given by the formula:
• B = (μ₀I)/(2πr), where μ₀ is the permeability of free space.
• This expression is valid for an infinitely long straight conductor.
• The magnetic field forms concentric circles around the wire, decreasing with distance as 1/r.
• This is a key formula for JEE, NEET, and board exam problems involving magnetic fields due to straight conductors.
2. What is the magnetic field pattern due to a straight wire?
The magnetic field around a straight current-carrying wire forms concentric circular lines centered on the wire.
• The field lines are perpendicular to the wire at every point.
• The direction of these circles can be determined using the right-hand thumb rule: if the thumb points along the current, fingers curl in magnetic field direction.
• The magnetic field is stronger closer to the wire and weakens as the distance increases.
3. How does the direction of the magnetic field depend on current direction in a straight wire?
The direction of the magnetic field around a straight wire depends on the current's direction and is determined using the right-hand thumb rule.
• Point your right thumb in the direction of current (I).
• The curled fingers show the direction of magnetic field lines around the wire.
• If current reverses, the field direction also reverses.
• This rule applies for both vertical and horizontal wire arrangements.
4. What is the difference between infinite and finite wire formulas for magnetic field?
The formula for the magnetic field due to a wire varies with its length relative to the observation point:
• Infinite wire: B = (μ₀I)/(2πr) (valid when the wire is much longer than the distance from the point)
• Finite wire: B = (μ₀I)/(4πr) × (sinθ₁ + sinθ₂), where θ₁ and θ₂ are the angles from the point to the ends of the wire.
• Use the infinite wire formula only when the observation point is far from the wire ends; otherwise, use the finite wire expression in calculations and derivations.
5. Does a straight wire create a magnetic field if current flows through it?
Yes, any straight wire carrying an electric current produces a magnetic field around it.
• The field strength depends on current (I) and distance from the wire (r).
• The direction is given by the right-hand thumb rule.
• This phenomenon is fundamental and applies to all conductors—discovered by Oersted's experiment.
6. How can you experimentally observe the magnetic field due to a straight current-carrying wire?
You can observe the magnetic effect using a simple experiment with a compass and a straight wire:
1. Place a straight wire over a magnetic compass.
2. Pass current through the wire.
3. The compass needle deflects, indicating the presence and direction of a magnetic field.
4. Sprinkling iron filings around the wire shows concentric magnetic field patterns.
This is a classic demonstration of Oersted's experiment, required knowledge for class 10 and 12 syllabus.
7. What is the right-hand thumb rule for magnetic field around a straight conductor?
The right-hand thumb rule helps determine the direction of the magnetic field:
• Point your right-hand thumb in the direction of the electric current (I).
• The direction in which your fingers curl gives the direction of magnetic field lines (B) encircling the wire.
• This visual technique is essential for solving exam problems involving field direction.
8. When should you use the Biot-Savart law versus Ampere's Law for finding the magnetic field due to a straight wire?
Use the Biot-Savart law for arbitrary wire shapes or when calculating the magnetic field at any distance and orientation, and Ampere’s law for infinite, symmetric conductors.
• For an infinite straight wire, both laws give the same result, but Ampere’s law is quicker due to symmetry.
• For a finite or semi-infinite straight wire, use Biot-Savart law to account for end effects and angle dependence.
• Always check the problem’s symmetry before choosing the method.
9. Can the magnetic field inside a thick wire be different from the field just outside the wire?
Yes, the magnetic field inside a current-carrying wire increases linearly with distance from the center, while outside it follows the 1/r law.
• Inside (at radius x < wire radius R): B = (μ₀Ix)/(2πR²)
• Outside (x > R): B = (μ₀I)/(2πx)
• This distinction is critical in advanced problems and real conductors.
10. What mistakes should students avoid when applying the magnetic field formula for a straight wire?
Common mistakes to avoid include:
• Forgetting to include 2π in the denominator (B = μ₀I/2πr)
• Confusing the directions—always use the right-hand thumb rule for clarity.
• Using the infinite wire formula for short or finite wires—use Biot-Savart for these cases.
• Not converting units (current in amps, distance in meters, μ₀ in T·m/A)
• Missing the field’s 1/r dependency—don't assume the field is uniform at all points.
