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Question : 21
Total: 37
SECTION - B
Derive the expression for the torque acting on an electric dipole, when it is held in a uniform electric field. Identify the orientation of the dipole in the electric field, in which it attains a stable equilibrium.
OR
Obtain the expression for the energy stored in a capacitor connected across a d.c. battery. Hence, define energy density of the capacitor.
Solution:
An electric dipole AB consisting of charge + q and − q and of length 2 a is placed in uniform electric field E making an angle θ with the direction of electric field.
Force acting on− q is − q E
Force acting on+ q is q E
These two forces are equal and opposite to each other. Hence, a torque on the dipole is developed.
Torque= Force × perpendicular distance between the forces
Or, τ = q E × 2 a s i n θ
Or, τ = ( q × 2 a ) E s i n θ
∴ τ = PE s i n θ
∴ τ = PE s i n θ (where P is dipole moment)
Dipole will attain stable equilibrium when it will be oriented along the direction of electric field.
OR
A capacitor is connected across the terminals of a d.c. battery.
The energy stored on a capacitor is equal to the work done by the battery.
Work done to move a small amount of charge dQ from the negative plate to the positive plate of the capacitor is equal toV dQ , where V is the voltage across the capacitor.
dU = VdQ =
dQ
∴ Energy stored =
U = ∫ VdQ =
∫ QdQ =
=
CV 2 ......(i)
Energy density is defined as the total energy per unit volume of the capacitor.
For a parallel plate capacitor,
C =
Putting in eqn (i)
U =
V 2 =
Ad (
) 2
U =
Ad 2 ( putting
= E )
A× d = Volume of space between plates
So, energy stored per unit volume
Force acting on
Force acting on
These two forces are equal and opposite to each other. Hence, a torque on the dipole is developed.
Torque
Dipole will attain stable equilibrium when it will be oriented along the direction of electric field.
OR
A capacitor is connected across the terminals of a d.c. battery.
The energy stored on a capacitor is equal to the work done by the battery.
Work done to move a small amount of charge dQ from the negative plate to the positive plate of the capacitor is equal to
Energy density is defined as the total energy per unit volume of the capacitor.
For a parallel plate capacitor,
Putting in eqn (i)
A
So, energy stored per unit volume
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