Question

Explain why Ampere's law for magnetism needs to be adjusted to include a "displacement current" term and what that implies about changing electric field

Answer 1

We know that a magnetic field is produced by a moving charged particle. In order to visualize how magnetic field is produced due to a change in the electric field, we consider a capacitor of capacitance C, that is being charged and the current flowing through the capacitor is time dependent and is given by i(t).

Applying the Ampere’s circuital law to the system, we can write,

This equation gives the magnetic field at any point outside a capacitor. In order to find the magnetic field at any point A outside the capacitor, we consider a plane circular loop such that its radius is r and its place is perpendicular to the wire connected to this capacitor. Here, the magnetic field is uniform and its magnitude is the same at all the points in the circular loop under consideration. We can write,

  

Now, we consider another case in which a semi-capsule shaped surface is considered which has its rim between the two capacitor plates.

Now, as we apply the Ampere’s Law to point A and B, we notice that, at point A the equation is same as the equation written above, but at point B of the equation becomes zero, as no current passes through those surfaces. So, we can say that from case one there is a magnetic field at the point A but from the second case, the magnetic field at the point A is zero. This case cannot be explained until we reconsider the Ampere’s Circuital Law. Upon further analysis it can be observed that there was a term missing in the equation representing the Ampere’s circuital law. Now taking the electric field into consideration.

For a capacitor with plate area A and a charge Q, the magnitude of electric field between the plates of the capacitor can be given as,

  

And as per the Gauss’s law, the flux through the surface is given by,

Now as we change the charge on the capacitor, after charging with a current i = (dQ/dt), we can write,

The above equations are consistent, if and only if,

The above mentioned term was the missing term in the Ampere’s Circuital law. We thus generalize this term and add it to the law, which leads us to the same value of the magnetic field in all situations.

As we know, the current caused due to the flow of charges in a conductor is termed as the conduction current. The current due to the change in the electric field, as defined by the above equation is a term different from conduction current. It is termed as the displacement current.

We can thus write, the total current passing through the capacitor can be given as,

  

Here, i_c is the conduction current and i_d is the displacement current. The generalized Ampere’s circuital law is given by,

  

The above equation is termed as the Ampere-Maxwell law.Now changing this equation by differentional form,

  

Hence it implies that a changing electric field will produce a magnetic field.