Magnetic Field Inside and Outside: Formulas, Graphs, and Key Differences
The magnetic field due to infinite wire in cylinder is a classic case of symmetry in electromagnetism, vital for JEE Main Physics. Using Ampère’s law, you can easily calculate the magnetic field both inside and outside a straight, infinite current-carrying wire or a cylindrical conductor. This topic helps decode how current distribution and geometry change the observed magnetic field in and around wires, solid cylinders, and hollow (tubular) cylinders.
Consider a long wire or cylinder with radius R carrying a steady current I. The point of interest is usually at a distance r from the axis, where r may be less than, equal to, or greater than R. Understanding which formula to apply for each case, visualizing B versus r, and recognizing application traps are key JEE skills.
Magnetic Field Due to Infinite Wire in Cylinder: Key Formulas and Summary
| Location | Formula for B | Condition |
|---|---|---|
| Outside infinite wire/cylinder | B = μ₀I / 2πr | r > R |
| Inside solid cylinder | B = μ₀Ir / 2πR² | r < R |
| Inside hollow cylinder | B = 0 | r < inner radius |
| Within thickness of hollow cylinder | B ∝ (r² - a²) | a < r < b |
For an infinite straight conductor, use B = μ₀I / 2πr outside the wire regardless of cylinder or wire, as long as r > R. For points inside a solid cylindrical wire (r < R), apply B = μ₀Ir / 2πR². This shows B varies linearly with r inside, peaking at the surface.
Stepwise Derivation & Conceptual Approach for Magnetic Field Due to Infinite Wire in Cylinder
- Apply Ampère’s law: ∮B⋅dl = μ₀Ienc.
- Choose a circular Amperian loop of radius r centered on the axis.
- For r < R (inside solid cylinder), only current through area (πr²) is enclosed.
- Get Ienc = I (r²/R²) using current density.
- Solve for B: B·2πr = μ₀I(r²/R²).
- This yields B = μ₀Ir / 2πR² for inside.
- For r > R (outside), Ienc = I. So B·2πr = μ₀I, giving B = μ₀I / 2πr.
If the cylinder is hollow (current only in the cylindrical shell), within the cavity (r less than inner radius), B = 0. However, within the shell’s thickness, B varies based on how much current is enclosed, calculated using area inside the Amperian loop.
For deeper insight, you can review the Biot Savart law page for the fundamental law, and see magnetic field due to straight wire explained with visualizations.
Variation, Graphs, and Magnetic Field Direction for Infinite Wire and Cylinder
The magnetic field due to infinite wire in cylinder has a predictable change with radius:
- B increases linearly from the axis to the surface (r = 0 to r = R) in solid cylinders
- B drops with 1/r outside the cylinder (r > R)
- B is zero within the hollow cavity
- Direction is always tangential, following the right-hand rule
If you draw B versus r, the graph rises straight from zero to a max for r = R (inside solid), then falls hyperbolically (1/r) outside. For a hollow cylinder, field is zero in the cavity, then spikes at the shell.
See also magnetic effects of current and magnetism for topic-wide application, and right hand thumb rule for direction checks.
Comparing Solid vs Hollow Cylinder and Practical JEE Applications
| Aspect | Solid Cylinder | Hollow Cylinder |
|---|---|---|
| B inside (cavity) | Linearly increases | Zero |
| B in shell region | N/A | Increases then decreases |
| B outside | 1/r law | 1/r law |
Solid conductors distribute current over their total cross section, while in hollow cylinders current flows in the shell’s thickness. Watch out for confusion between distance r from axis and radius R of the cylinder.
- Problems may specify “inside the wire/cylinder” or “outside,” which determines formula usage
- Use ampere law page for derivation logic
- Practice MCQs on magnetic field due to toroid for geometry variety
When solving JEE numericals, always:
- Check if the point is inside, within the shell, or outside
- Plug correct values—especially “r” in the formula
- Use SI units: B in tesla, I in ampere, r, R in metre
- Apply direction with the right hand thumb rule
- For comparison questions, build a mini-table as shown above
See properties of diamagnetic paramagnetic and ferromagnetic materials to explore material effects, and current electricity basics for current density pitfalls.
Example – Numerical Application for Magnetic Field Due to Infinite Wire in Cylinder
A long solid cylinder, radius 2.0 cm, carries current 8.0 A uniformly. What is the magnetic field at:
- r = 1.0 cm (inside)
- r = 2.0 cm (surface)
- r = 3.0 cm (outside)
Stepwise solution:
- Inside (r = 1.0 cm):
B = μ₀Ir / 2πR² = (4π × 10⁻⁷ × 8 × 0.01) / (2π × 0.02²) = 0.8 × 10⁻⁴ T - On Surface (r = 2.0 cm):
B = μ₀I / 2πR = (4π × 10⁻⁷ × 8) / (2π × 0.02) = 8 × 10⁻⁵ T - Outside (r = 3.0 cm):
B = μ₀I / 2πr = (4π × 10⁻⁷ × 8) / (2π × 0.03) ≈ 5.3 × 10⁻⁵ T
This clear step-by-step breakdown matches JEE standards and builds problem confidence. To boost your practice, see current electricity mock test and magnetic effects of current and magnetism mock test 2.
Experimentally, the field can be mapped by placing a small compass or iron filings along various points outside the wire or cylinder. The field direction follows concentric circles around the wire axis. Review more on practical setups at experimental skills revision notes.
- For field in loops, see magnetic field on the axis of a circular current loop
- For direction rules, refer to magnetic lines of force
- For in-depth law, check gauss law in magnetism
- For induction insights, see electromagnetic induction and alternating current revision notes
- For comparison of series-parallel conductors, visit difference between series and parallel circuits
Always remember, the magnetic field due to infinite wire in cylinder is best mastered by layering formula memory, graphical intuition, and problem solving. Keep symbols clear—B for field, I for current, r for distance from axis, μ₀ for magnetic constant (4π × 10⁻⁷ T·m/A). For instant revision before exams, Vedantu has high-yield revision notes and concept maps tailored for JEE aspirants.
FAQs on Magnetic Field Due to an Infinite Wire Inside a Cylinder
1. What is the formula for the magnetic field due to an infinite straight wire?
The standard formula for the magnetic field at a distance r from an infinite straight current-carrying wire is B = (μ₀I) / (2πr).
- μ₀ is the permeability of free space
- I is the current through the wire
- r is the perpendicular distance from the wire
2. What is the formula for the magnetic field inside a solid cylinder carrying current?
Inside a solid cylindrical conductor, the magnetic field at a radial distance r from the axis (for r < R, where R is cylinder radius) is B = (μ₀I r) / (2πR²).
- This field varies linearly with r inside the cylinder
- Maximum value at surface (r=R); use in questions about field variation within conductors
3. How do you calculate the magnetic field inside a hollow cylinder (cylindrical shell)?
For a hollow cylinder (cylindrical shell), the magnetic field depends on the location:
- Inside the hollow part (r < inner radius): B = 0
- Within conductor thickness: Use Ampere's law considering current enclosed
- Outside the cylinder (r > outer radius): B = (μ₀I) / (2πr)
4. How does the magnetic field vary with distance from the axis in an infinite cylinder?
The magnetic field in an infinite current-carrying cylinder changes with radial distance (r):
- Inside (r < R): B increases linearly with r (B ∝ r)
- At the surface (r = R): Maximum field
- Outside (r > R): B decreases as 1/r
5. What is the difference between the magnetic field inside a wire and inside a hollow cylinder?
The difference relies on current distribution:
- Solid wire: Field inside increases with distance (B ∝ r), not zero at inner points
- Hollow cylinder: Field is zero inside the hollow part (no current enclosed)
6. How do you derive the formula for the magnetic field inside and outside a cylinder using Ampère's Law?
To derive these formulas, apply Ampère’s Law with a circular Amperian loop:
- Inside (r < R): Integrate current density over enclosed area, leading to B = (μ₀Ir)/(2πR²)
- Outside (r > R): Take total current I, giving B = (μ₀I)/(2πr)
7. How do you draw the magnetic field versus radial distance graph for an infinite wire in a cylinder?
The B versus r graph for an infinite wire/cylinder is:
- Linear increase for r < R (inside conductor): B ∝ r
- Decreases as 1/r for r > R (outside): B ∝ 1/r
- B at r = 0 is zero (center)
8. What experiment can demonstrate the magnetic field due to a current-carrying wire or cylinder?
You can demonstrate this with a simple lab setup:
- Wrap a wire around a cylinder and connect it to a battery (steady current)
- Use a small magnetic compass or iron filings to trace magnetic field lines around it
- Observe concentric circles of the field, matching theory from Ampère’s Law
9. Is the magnetic field zero at the center of a solid cylinder carrying current?
Yes, at the exact center (r = 0) of a solid current-carrying cylinder, the magnetic field B is zero.
- This is because no current is enclosed by the Amperian loop at the center
- Field increases linearly as you move away from the center
10. What are common mistakes students make when solving magnetic field problems for wires and cylinders?
Common mistakes include:
- Confusing r (distance from axis) with R (radius of cylinder)
- Using wrong formulas for hollow vs solid conductors
- Ignoring whether the point is inside or outside
- Forgetting units (e.g., tesla for B)
11. Does current density affect the magnetic field inside a cylinder?
Yes, the current density (J) directly affects the magnetic field inside a solid cylinder:
- If current is uniformly distributed, use B = (μ₀J r)/2
- Non-uniform current changes the field's mathematical form
12. Are the magnetic field formulas the same for AC and DC currents in a cylinder?
For steady (DC) currents, use the standard formulas. For AC, the magnetic field at each instant is the same as for DC, but B varies with time as the current oscillates.
- Additional factors like electromagnetic induction may appear at high frequencies
- For basic problems, use standard DC formulas unless specified
