Derivations: Displacement Current, Magnetic Intensity
& Magnetisation, Moving Coil Galvanometer
NCERT Class XII Physics
1 Displacement Current
¸
⃗ · d⃗l = µ0 Ic fails while charging a capacitor: for a plane surface cutting the wire,
Ampere’s law B
Ic = I; for a surface bulging through the gap between the plates, Ic = 0. Same loop, different
answers — contradiction.
q
q
Between the plates (area A, charge q): E =
, so electric flux ΦE = EA = .
ε0 A
ε0
dΦE
1 dq
Ic
=
=
dt
ε0 dt
ε0
Maxwell defined the displacement current:
Id = ε0
dΦE
dt
which equals Ic , keeping current continuous. Ampere’s law is generalised (Ampere–Maxwell Law):
˛
⃗ · d⃗l = µ0 (Ic + Id )
B
2 Magnetic Intensity H
H is the field due to free (external) current alone:
H=
B0
µ0
For a solenoid (N turns, length l, current I): B0 = µ0 nI, so
H = nI =
NI
l
3 Magnetisation M
Net magnetic moment per unit volume of the material:
M=
m
V
For most materials M ∝ H, giving the susceptibility χm :
M = χm H
1
4 Total Field B and Permeability
B = field from free current + field from magnetisation:
B = µ0 H + µ0 M
B = µ0 (H + M )
Using M = χm H: B = µ0 (1 + χm )H = µ0 µr H, so
µ r = 1 + χm
5 Moving Coil Galvanometer
Coil: N turns, area A, current I, in radial field B (torque independent of orientation, sin θ = 1
always).
Deflecting torque = N IAB. Restoring torque of spring = kϕ. At equilibrium:
N IAB = kϕ
I=
k
N AB
Is =
N AB
ϕ
=
I
k
ϕ
Current sensitivity:
Voltage sensitivity (R = coil resistance, V = IR):
Vs =
ϕ
N AB
=
V
kR
Note: increasing N raises Is but also raises R ∝ N , so Vs stays unchanged.
2