can an Amperian loop ever have a non-zero value of its line integral with the magnetic field.
no
Yes, an Amperian loop can have a non-zero value of its line integral with the magnetic field. The line integral of the magnetic field around a closed loop is directly related to the net current passing through the loop. This relationship is given by Ampere's law.
Ampere's law states that the line integral of the magnetic field (B) around a closed loop (C) is equal to the product of the permeability of free space (μ₀) and the net current (I_enc) passing through the loop. Mathematically, it is written as:
∮B·dl = μ₀I_enc
In this equation, ∮B·dl represents the line integral of the magnetic field around the loop, dl represents a small vector element along the loop's path, and μ₀ is a constant known as the permeability of free space.
If the net current passing through the loop is non-zero, then the line integral of the magnetic field will also be non-zero. This can occur, for example, if there is a current-carrying wire passing through the loop.
To determine the line integral of the magnetic field around an Amperian loop, you need to know the value and direction of the magnetic field at all points along the loop and calculate the line integral using appropriate mathematical techniques like integrating along the path of the loop.
In summary, an Amperian loop can have a non-zero value of its line integral with the magnetic field if there is a net current passing through the loop. The line integral is determined by Ampere's law, which relates the magnetic field and the net current passing through the loop.