Maxwell’s correction to Ampere’s law
- Ampere’s law states that “the line integral of resultant magnetic field along a closed plane curve is equal to μ0 time the total current crossing the area bounded by the closed curve provided the electric field inside the loop remains constant".
- Ampere’s law is true only for steady currents.
- Maxwell found the shortcoming in Ampere’s law and he modified Ampere’s law to include time-varying electric fields.
- For Ampere’s circuital law to be correct Maxwell assumed that there hasto be some current existing between the plates of the capacitor.
- Outside the capacitor current was due to the flow of electrons.
- There was no conduction of charges between the plates of the capacitor.
- According to Maxwell between the plates of the capacitor there is an electric field which is directed from positive plate to the negative plate.
- Magnitude of the electric field E =(V/d)
- Where V=potential difference between the plates, d = distance between the plates.
- E = (Q/Cd)
- where Q=charge on the plates of the capacitor,Capacitance of the capacitor=C
- =>= (Q/ (Aε0d/d))where A =area of the capacitor.
- Direction of the electric field will be perpendicular to the selected surface i.e. if considering plate of the capacitor as surface.
- As E =0 outside the plates and E=(Q/(Aε0)) between the plates.
- There may be some electric field between the plates because of which some current is present between the plates of the capacitor.
- Electric Flux through the surface=ΦE= (EA) =(QA)/ (Aε0) =(Q/ ε0)
- Assuming Q (charge on capacitor i.e. charging or discharging of the capacitor) changes with time current will be get generated.
- Therefore current Id =(dQ/dt)
- Where Id =displacement current
- =>DifferentiatingΦE =(Q/ ε0) on both sides w.r.t time,
- (dΦE/dt) =(1/ ε0) (dQ/dt)
- Therefore (dQ/dt) = ε0 (d ΦE/dt)
- =>Current was generated because of change of electric flux with time.
- Electric flux arose because of presence of electric field in the plates of the capacitor.
- Id = (dQ/dt) = Displacement current
- Therefore Change in electric field gave rise to Displacement current.
- Current won’t be 0 it will be Id.
- There is some current between the plates of the capacitor and there is some current at the surface.
- At certain points there is no displacement current there is only conduction current and vice-versa.
- Maxwell corrected the Ampere’s circuital law by including displacement current.
- He said that there is not only the current existed outside the capacitor but also current known as displacement currentexisted between the plates of the capacitor.
- Displacement current exists due to the change in the electric field between the plates of the capacitor.
- Conclusion:-Magnetic fields are produced both by conduction currents and by time varying fields.
Problem:- A parallel plate capacitor (Fig) made of circular plates each of radiusR = 6.0 cm has a capacitance C = 100 pF. The capacitor is connected toa 230 V ac supply with a (angular) frequency of 300 rad s–1.
(a) What is the rms value of the conduction current?
(b) Is the conduction current equal to the displacement current?
(c) Determine the amplitude of B at a point 3.0 cm from the axisbetween the plates.
Answer:- Radius of each circular plate, R = 6.0 cm = 0.06 m
Capacitance of a parallel plate capacitor, C = 100 pF = 100 × 10−12 F
Supply voltage, V = 230 V
Angular frequency, ω = 300 rad s−1
(a) Rms value of conduction current, I
XC = Capacitive reactance
Therefore, I = V × ωC
= 230 × 300 × 100 × 10−12
= 6.9 × 10−6 A
= 6.9 μA
Hence, the rms value of conduction current is 6.9 μA.
(b) Yes, conduction current is equal to displacement current.
(c) Magnetic field is given as:
B = (μ0r)/ (2 R2) I0 Where,
μ0 = Free space permeability = 4x πx10-7NA-2
I0 = Maximum value of current =√2 I
r = Distance between the plates from the axis = 3.0 cm = 0.03 m
Therefore B = (4xπ x10-7x0.03x√2x6.9x10-6)/ (2xπ x0.06)2
= 1.63 × 10−11 T
Hence, the magnetic field at that point is 1.63 × 10−11 T.