Inconsistency of Ampere's Circuital Law
Examines the failure of Ampere's circuital law to hold true in situations involving time-varying electric fields, such as a charging capacitor.
What is Inconsistency of Ampere's Circuital Law?
Electric current due to the flow of charge carriers (e.g., electrons in a wire).
Key formula / rule: Ampere's Circuital Law (Original)
Key points
- Understand the limitations of Ampere's circuital law.
- Identify situations where Ampere's law is inconsistent.
- Appreciate the need for Maxwell's displacement current.
Common exam trap
Assuming Ampere's law is valid for all situations, including changing electric fields.
Definitions
- Term
Conduction Current
- Meaning
Electric current due to the flow of charge carriers (e.g., electrons in a wire).
- Term
Displacement Current
- Meaning
A current-like term introduced by Maxwell, proportional to the rate of change of electric flux, which can also produce a magnetic field.
- Term
Amperian Loop
- Meaning
An imaginary closed loop used in Ampere's circuital law to calculate the magnetic field.
Learning objectives
Understand the limitations of Ampere's circuital law.
Identify situations where Ampere's law is inconsistent.
Appreciate the need for Maxwell's displacement current.
Formulae
- Name
Ampere's Circuital Law (Original)
- Note
Valid only for steady currents and static fields.
- Expression
\\oint \\vec{B} \\· d\\vec{l} = \\μ0 Ienc
- Name
Electric Flux
- Note
Used in the definition of displacement current.
- Expression
\\ΦE = \\int \\vec{E} \\· d\\vec{A}
- Name
Displacement Current
- Note
Introduced by Maxwell to account for changing electric fields.
- Expression
ID = \\ε0 \\frac{d\\ΦE}{dt}
- Name
Ampere-Maxwell Law
- Note
Modified Ampere's law, valid for time-varying fields.
- Expression
\\oint \\vec{B} \\· d\\vec{l} = \\μ0 (Ic + ID) = \\μ0 Ic + \\μ0 \\ε0 \\frac{d\\ΦE}{dt}
Prerequisites
Ampere's Circuital Law
Magnetic field due to current
Electric flux
Capacitance and charging of capacitors
Common mistakes
Assuming Ampere's law is valid for all situations, including changing electric fields.
Not considering the displacement current when analyzing circuits with capacitors.
Confusing conduction current with displacement current.
Keywords
Ampere's Law
Displacement Current
Maxwell
Charging Capacitor
Electromagnetic Induction
Time-varying fields
Ampere-Maxwell Law
Practice preview
During the charging of a parallel plate capacitor, Maxwell modified Ampere's Circuital Law by introducing the concept of:…
easy
Which of the following scenarios would NOT exhibit an inconsistency with the original Ampere's Circuital Law (∮ B ⋅ dl = μ₀ Ic)?…
medium
The modified Ampere's Circuital Law, including Maxwell's contribution, is given by ∮ B ⋅ dl = μ₀(Ic + Id). In the case of a charging capacitor, which term is responsible for the inconsistency of the original law?…
medium