Ampere’s Law gives the relationship between the magnetic field around a closed loop and the current enclosed by that loop.
Ampere’s Law Statement:
The line integral of the magnetic field B around a closed path is equal to μ₀ times the total current I enclosed by the path.
Mathematically:
∮ B · dl = μ₀ × I_enclosed
Where:
- ∮ B · dl = Line integral of the magnetic field along the closed path
- I_enclosed = Net current enclosed by the path
- μ₀ = Permeability of free space = 4π × 10⁻⁷ T·m/A
Application of Ampere’s Law: Magnetic Field of Symmetrical Configurations
Ampere’s Law is most useful when high symmetry is present.
1. Infinite Straight Wire
B = (μ₀ × I) / (2π × r) - (Same result as Biot–Savart Law)
2. Long Solenoid
B = μ₀ × n × I
Where:
- n = Number of turns per unit length
- I = Current through the solenoid
3. Toroid
B = (μ₀ × N × I) / (2π × r)
Where:
- N = Total number of turns
- r = Distance from the center of the toroid
Biot–Savart Law vs Ampere’s Circuital Law
| Feature | Biot–Savart Law | Ampere’s Law |
|---|---|---|
| Applicable to | Any current configuration | Symmetrical current configurations |
| Type of equation | Differential form | Integral form |
| Use | To calculate B at a point | To calculate B along a closed path |
| Best for | Arbitrary shapes | Highly symmetrical cases |
Applications
- Used in the design and analysis of electrical machines (motors, transformers)
- Essential for electromagnetic field calculations
- Basis for magnetic field sensors and inductive systems
- Ampere’s Law is heavily used in solenoids, coaxial cables, and toroidal inductors
Frequently Asked Questions (FAQs) on Ampere’s Circuital Law
Q1. What is Ampere’s Circuital Law?
- Ampere’s Circuital Law states that the line integral of the magnetic field B around a closed loop is equal to the product of the permeability of free space μ₀ and the total current I enclosed by the loop.
Q2. What is the significance of Ampere’s Law?
- Ampere’s Law is a powerful tool for calculating the magnetic field in symmetrical current distributions like infinite wires, solenoids, and toroids. It simplifies complex calculations compared to Biot–Savart Law in such cases.
Q3. What is the value of μ₀ (mu naught)?
- The value of μ₀ (permeability of free space) is 4π × 10⁻⁷ T·m/A (Tesla meter per ampere).
Q4. How is Ampere’s Law different from Biot–Savart Law?
- Ampere’s Law is an integral form used mainly for symmetrical setups.
- Biot–Savart Law is a general law used for any shape of current-carrying conductor.
- Ampere’s Law gives quicker results when symmetry is present, such as in solenoids or toroids.
Q5. What are the applications of Ampere’s Law?
Ampere’s Law is used to calculate magnetic fields in:
- Long straight conductors
- Solenoids
- Toroids
- Coaxial cables
Q6. Can Ampere’s Law be applied to all current-carrying conductors?
- No, Ampere’s Law is best applied where symmetry exists (cylindrical, planar, or toroidal). In irregular or asymmetric current paths, Biot–Savart Law or numerical methods are more suitable.
Q7. What is the magnetic field inside a long solenoid using Ampere’s Law?
For a long solenoid:
B = μ₀ × n × I,
where n is the number of turns per unit length and I is the current.
Q8. Is Ampere’s Law part of Maxwell’s Equations?
- Yes, Ampere’s Law (with Maxwell's correction) is one of the four Maxwell’s Equations and describes how electric currents and changing electric fields produce magnetic fields.