CBSE Class 12-science Answered

A parallel plate capacitor is being charged by a time varying current. Explain brieflyhow Ampere's circuital law is generalized to incorporate the effect due to the displacement current

Asked by eshoreabhijit | 02 Mar, 2018, 03:02: PM

Expert Answer

Figure(a) shows a circular plate capacitor charged by a current i through wire. Current enters the left-side plate and leaves the right-side plate. An amperian loop surrounds the wire in figure(a) and forms the boundary for a surface that is pierced by the wire. The current in the wire sets up a magnetic field. Magnetic field and current are related by Ampere's law as given below

begin mathsize 12px style contour integral bold italic B times bold italic d bold italic l space equals space mu subscript 0 space i end style

..............................(1)

Line integral of the magnetic field around the loop is proportional to the total current that passes through the surface bound by the loop.

In Figure(b) we have kept the same loop but stretched the surface so that it encloses the left-side capacitor plate. Since the loop has not changed the left side of equation(1) remains same for figure (b). But right side of equation(1) vanishes in this case because there is no conduction current.

To restore the ampere's law so that it also explains the situation shown in figure(b), we use the physics: a magnetic field is set up by a changing electric field.

Let us consider the situation shown in figure(b) in more detail. As the charges transported into the capacitor, electric field in interior of the capacitor is changing at certain rate dE/dt. Since the electric field lines are passing through the surface of amperian loop, we account for this passage of electric field lines in terms of electric flux. Changing electric field must give a corresponding change in electric flux dΦ_E/dt.

Faradays law states that electric field is set up by changing magnetic field. By symmmetry we write now

begin mathsize 12px style contour integral bold italic B times bold italic d bold italic l space equals space mu subscript 0 space epsilon subscript 0 fraction numerator d ϕ subscript E over denominator d t end fraction end style

.....................(2)

Equation(2) is mathematical form of saying that magnetic field is set up by changing electric field. μ₀ and ε₀ are appearing for the sake of SI unit

Equation(1) describes the situation shown in figure(a), where as Equation (2) describes the situation shown in figure (b). In first cae it is the current through the surface of amperian loop that gives the magnetic field. In second case the magnetic field is set up by changing electric field. To generalise both the cases, we have to modify the ampere law and we write