0.10 Ampere's law  (Page 4/6)

This fact underlines an important fact that absence of current across Ampere loop does not ensures that magnetic field in a region is zero. We can verify this by using a square loop inside a solenoid. A solenoid, as we shall study, produces a uniform magnetic field within it. Let the magnetic field be B as shown. Clearly, there is no current passing through the enclosure of the square loop as current in solenoid flows through the helical coil covering the region under consideration. Let us now carry out the integration in clockwise direction along ACDEA.

Magnetic field at a point

$$\oint \mathbf{B}.đ\mathbf{l}={\int }_{AC}\mathbf{B}.đ\mathbf{l}+{\int }_{CD}\mathbf{B}.đ\mathbf{l}+{\int }_{DE}\mathbf{B}.đ\mathbf{l}+{\int }_{EA}\mathbf{B}.đ\mathbf{l}$$ $$\oint \mathbf{B}.đ\mathbf{l}={\int }_{AC}Bđl\cos 0°+{\int }_{CD}Bđl\cos 90°+{\int }_{DE}Bđl\cos 180°+{\int }_{EA}Bđl\cos 90°$$ $$\oint \mathbf{B}.đ\mathbf{l}=Ba+0-Ba+0=0$$

Clearly, existence of magnetic field does not require net current through the loop. For another example, see Ampere's law(exercise) : Problem 3

Maxwell modification

The basic assertion of Maxwell electromagnetic theory is that changing electric filed sets up magnetic field in the same manner in which a varying magnetic field sets up electric field as given by Farady’s induction law. The Maxwell equation is complementary to Farady’s induction law and is given as :

$$\oint \mathbf{B}.đ\mathbf{l}={\mu }_0{\epsilon }_0\frac{đ{\phi }_E}{đt}$$

Note how the time rate of change of electric field $\frac{đ{\phi }_E}{đt}$ is related to magnetic field ( B ) by this equation. In order to account for this additional cause of magnetic field resulting from varying electric field, a more generalized form of Ampere law including the term given by Maxwell is :

$$\oint \mathbf{B}.đ\mathbf{l}={\mu }_{0}I+{\mu }_0{\epsilon }_0\frac{đ{\phi }_E}{đt}$$

Of course for situation involving only steady current, the form of Ampere’s law is reduced to its original form.

The presence of magnetic field between capacitor plates during charging of a capacitor confirms Maxwell law. As the charge builds up on the capacitor plate, there is varying electric field in the gap between plates. This varying electric field, in turn, sets up magnetic field. We can, therefore, suggest that the varying electric field is equivalent to a current. After all, a current also produces magnetic field. But we know there is no actual current between two plates. Hence, this equivalent current is a sort of pseudo current and is known as "displacement" current, which when present would have produced the same magnetic field in the gap as actually produced by the varying electric field.

We should understand that this assertion about displacement current or setting up of magnetic field due to varying electric field is an important step in explaining electromagnetic propagation. In a nutshell, it says that the presence or propagation of magnetic or electric field do not require either a charge or a current. That is exactly what we see with the propagation of electromagnetic field which is known to be composed of time varying electric and magnetic components. The changing electric field sets up magnetic field and changing magnetic field sets up electric field in a complementary manner. This is how electromagnetic field is continuously driven to propagate electromagnetic wave without presence of either charge or current. In other words, the two varying fields drive each other without the conventional source like charge or current.