Short Answer
The Ampere-Maxwell law states that a magnetic field is produced not only by an electric current but also by a changing electric field. Maxwell added the idea of displacement current to Ampere’s original law to make it complete.
In mathematical form:
∮B · dl = μ₀ (I + ε₀ dΦₑ/dt)
This law strengthens the link between electricity and magnetism and helps in explaining how electromagnetic waves are formed.
Detailed Explanation :
Ampere-Maxwell Law
The Ampere-Maxwell law is one of the four Maxwell’s equations and plays a key role in connecting electricity and magnetism. Originally, André-Marie Ampère discovered that electric currents create magnetic fields. His law explained the magnetic field around a straight wire or around coils carrying current. However, Ampere’s law had a limitation: it did not apply correctly in situations where electric fields changed with time.
To fix this problem, James Clerk Maxwell introduced an important addition called displacement current, which allowed the law to work in all conditions. The combination of Ampère’s original idea and Maxwell’s correction is known as the Ampere-Maxwell law.
Statement of the Law
The Ampere-Maxwell law states that:
“A magnetic field is produced by both electric current and a changing electric field.”
Maxwell included the term ε₀ dΦₑ/dt, which represents the changing electric flux, to account for situations where no real current flows but the electric field is still changing. This completed the symmetry of Maxwell’s equations and helped explain electromagnetic waves.
Mathematically:
∮B · dl = μ₀ (I + ε₀ dΦₑ/dt)
Where:
- B = magnetic field
- dl = small length element of the loop
- I = electric current
- ε₀ dΦₑ/dt = displacement current
- μ₀ = permeability of free space
Ampère’s Original Law and Its Limitation
Ampère’s original law stated that the magnetic field around a closed loop depends on the electric current passing through that loop. It worked perfectly for steady (unchanging) currents. But it failed when currents varied with time, such as in capacitors.
For example, consider a capacitor charging in a circuit. Current flows into one plate and out of the other. Between the plates is an insulating material (or air) where no real current flows. According to Ampère’s original law, the magnetic field should change depending on how you choose your surface—this leads to contradiction.
This inconsistency required correction.
Maxwell’s Correction: Displacement Current
Maxwell introduced the idea of displacement current to fix the problem. He suggested that a changing electric field behaves like a current and can produce a magnetic field.
Displacement current is defined as:
I_d = ε₀ dΦₑ/dt
It is not an actual flow of electrons but a temporary current caused by the changing electric field. This allows Ampère’s law to hold true everywhere, even inside capacitors where real current cannot pass.
Significance of Displacement Current
Displacement current is one of Maxwell’s greatest contributions. It explains several important concepts:
- Magnetic fields can be created without real currents.
- A changing electric field produces a magnetic field.
- Electromagnetic waves can propagate even in a vacuum.
This correction brought perfect symmetry between electric and magnetic fields.
Role in Electromagnetic Waves
The Ampere-Maxwell law plays a central role in the formation of electromagnetic waves. Maxwell showed that:
- A changing electric field produces a magnetic field (Ampere-Maxwell law).
- A changing magnetic field produces an electric field (Faraday’s law).
These two laws together create a continuous cycle where electric and magnetic fields sustain each other. This leads to the formation of electromagnetic waves, such as light, radio waves, and X-rays. Because displacement current exists even in a vacuum, electromagnetic waves do not need any medium to travel.
Physical Interpretation
The Ampere-Maxwell law helps us understand:
- How currents and electric fields create magnetic fields
- Why magnetic fields exist between capacitor plates
- Why waves can exist in space
- How electric circuits behave when currents change
It provides a complete picture of magnetic field behaviour in both steady and changing conditions.
Application in Real Life
The Ampere-Maxwell law is important in many technologies:
- Capacitors in electronic circuits
- Antennas used for radio, TV, and mobile communication
- Transformers and inductors
- Electromagnetic wave transmission
- Power systems
- Wireless charging technologies
Without this law, modern electrical engineering would not have evolved.
Mathematical Form (Differential Form)
The Ampere-Maxwell law is also written in differential form:
∇ × B = μ₀J + μ₀ε₀ ∂E/∂t
Where:
- J is the current density
- ∂E/∂t represents change in electric field with time
This form is used in advanced physics and engineering to solve field problems.
Conclusion
The Ampere-Maxwell law states that magnetic fields are produced not only by electric currents but also by changing electric fields. Maxwell’s addition of displacement current completed Ampère’s law and created perfect harmony between electricity and magnetism. This law is essential for understanding how electromagnetic waves form and propagate, making it a cornerstone of modern physics and electrical technology.