Which of the following conditions is NOT a true derivative of Ampere’s circuit law? 

Explanation:

Ampere's Circuit Law

Definition: Ampere's circuit law, also known as Ampere's law, relates the circulating magnetic field in a closed loop to the electric current passing through the loop. It is one of Maxwell's equations and is fundamental in the study of electromagnetism. Mathematically, it is expressed as:

∮_L H · dl = I_enc

where:

• ∮_L H · dl is the line integral of the magnetic field H around a closed loop L.
• I_enc is the total current enclosed by the loop.

Correct Option Analysis:

The correct option is:

Option 1: ∇ × J = H

This option is not a true derivative of Ampere's circuit law. To understand why, let's delve deeper into the mathematical formulation and implications of Ampere's law:

Ampere's law in differential form is given by:

∇ × H = J

This equation states that the curl of the magnetic field H is equal to the current density J. In other words, the magnetic field around a current-carrying conductor is directly related to the current flowing through it. The operator ∇ × represents the curl, which measures the tendency of the field to circulate around a point.

In option 1, ∇ × J = H, the roles of H and J are incorrectly interchanged. The curl of the current density J does not yield the magnetic field H. Therefore, this expression is not consistent with Ampere's law and is incorrect.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 2: I_enc = ∫_S (∇ × H) · dS

This option is a valid expression derived from Ampere's law. By applying Stokes' theorem, which relates the line integral of a vector field over a closed loop to the surface integral of the curl of the vector field over a surface bounded by the loop, we get:

∮_L H · dl = ∫_S (∇ × H) · dS = I_enc

This shows that the total current enclosed by the loop is equal to the surface integral of the curl of the magnetic field over the surface bounded by the loop.

Option 3: I_enc = ∫_S J · dS

This option is also a valid expression. It states that the total current enclosed by the loop is equal to the surface integral of the current density J over the surface S. This is consistent with the definition of current density and the total current passing through a surface.

Option 4: I_enc = ∮_L H · dl

This option is the integral form of Ampere's law itself. It states that the line integral of the magnetic field H around a closed loop is equal to the total current I_enc enclosed by the loop. This is the fundamental form of Ampere's law and is correct.

Conclusion:

Understanding Ampere's circuit law and its derivatives is crucial in the study of electromagnetism. The law provides a relationship between the magnetic field and the electric current passing through a closed loop. The correct expression of this relationship is essential for accurately describing the behavior of electromagnetic fields. Option 1, ∇ × J = H, is not a true derivative of Ampere's law because it incorrectly interchanges the roles of the magnetic field H and the current density J.