1 Answer
Ampere's Circuital Law is a fundamental law of electromagnetism that relates the magnetic field around a closed loop to the electric current passing through the loop. In its integral form, the law states:
\[
\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}}
\]
where:
- \(\vec{B}\) is the magnetic field,
- \(d\vec{l}\) is the infinitesimal vector along the loop,
- \(\mu_0\) is the permeability of free space, and
- \(I_{\text{enc}}\) is the total current enclosed by the loop.
However, the law, in its classical form, has an **inconsistency** when applied to certain scenarios involving time-varying electric fields. The inconsistency arises because Ampere's law does not fully account for the magnetic field created by a **changing electric field**.
### The Inconsistency:
Ampere's law works perfectly when there is a steady current, but when the current is **changing with time**, it doesn't capture the effects caused by a time-varying electric field. This leads to the problem that the law doesn't hold in situations where there are changing electric fields (like in capacitors or other situations where the electric field is not constant).
### The Solution (Maxwell's Addition):
James Clerk Maxwell resolved this inconsistency by adding a term to Ampere’s law, which is now known as the **displacement current** term. The updated form of Ampere's law (with Maxwell’s correction) is:
\[
\oint \vec{B} \cdot d\vec{l} = \mu_0 \left( I_{\text{enc}} + \epsilon_0 \frac{d\Phi_E}{dt} \right)
\]
where:
- \(\epsilon_0\) is the permittivity of free space,
- \(\frac{d\Phi_E}{dt}\) is the rate of change of the electric flux through the surface bounded by the loop.
The additional term, \(\epsilon_0 \frac{d\Phi_E}{dt}\), accounts for the effect of **changing electric fields**. This corrected version of the law is valid for both steady and time-varying currents and is consistent with the behavior of electromagnetic waves.
### Summary:
- The inconsistency in Ampere’s law arises when dealing with **time-varying electric fields**.
- This was addressed by Maxwell, who added the **displacement current** term, making Ampere's law valid even when the electric field is changing with time.
This modification played a crucial role in the development of **electromagnetic theory**, leading to the realization that light and other electromagnetic waves are self-propagating disturbances in the electric and magnetic fields.
\[
\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}}
\]
where:
- \(\vec{B}\) is the magnetic field,
- \(d\vec{l}\) is the infinitesimal vector along the loop,
- \(\mu_0\) is the permeability of free space, and
- \(I_{\text{enc}}\) is the total current enclosed by the loop.
However, the law, in its classical form, has an **inconsistency** when applied to certain scenarios involving time-varying electric fields. The inconsistency arises because Ampere's law does not fully account for the magnetic field created by a **changing electric field**.
### The Inconsistency:
Ampere's law works perfectly when there is a steady current, but when the current is **changing with time**, it doesn't capture the effects caused by a time-varying electric field. This leads to the problem that the law doesn't hold in situations where there are changing electric fields (like in capacitors or other situations where the electric field is not constant).
### The Solution (Maxwell's Addition):
James Clerk Maxwell resolved this inconsistency by adding a term to Ampere’s law, which is now known as the **displacement current** term. The updated form of Ampere's law (with Maxwell’s correction) is:
\[
\oint \vec{B} \cdot d\vec{l} = \mu_0 \left( I_{\text{enc}} + \epsilon_0 \frac{d\Phi_E}{dt} \right)
\]
where:
- \(\epsilon_0\) is the permittivity of free space,
- \(\frac{d\Phi_E}{dt}\) is the rate of change of the electric flux through the surface bounded by the loop.
The additional term, \(\epsilon_0 \frac{d\Phi_E}{dt}\), accounts for the effect of **changing electric fields**. This corrected version of the law is valid for both steady and time-varying currents and is consistent with the behavior of electromagnetic waves.
### Summary:
- The inconsistency in Ampere’s law arises when dealing with **time-varying electric fields**.
- This was addressed by Maxwell, who added the **displacement current** term, making Ampere's law valid even when the electric field is changing with time.
This modification played a crucial role in the development of **electromagnetic theory**, leading to the realization that light and other electromagnetic waves are self-propagating disturbances in the electric and magnetic fields.