CBSE Class 12 Physics 2020 Delhi Set 1 Paper

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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 2a is placed in uniform electric field E making an angle θ with the direction of electric field.

Force acting on −q is −qE
Force acting on +q is qE
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=qE×2asin‌θ
‌ Or, ‌‌τ=(q×2a)Esin‌θ
∴‌τ=‌ PE ‌sin‌θ
∴‌‌τ=PE‌s‌i‌n‌θ‌‌‌ (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 to V‌dQ, where V is the voltage across the capacitor.
dU=VdQ=‌
Q
C
‌dQ

∴ Energy stored =
U=∫VdQ=‌
1
C
‌∫QdQ
=‌
1
2
‌
Q2
C
=‌
1
2
CV2
......(i)
Energy density is defined as the total energy per unit volume of the capacitor.
For a parallel plate capacitor,
C=‌
Aε0
d

Putting in eqn (i)
U=‌
1
2
‌
Aε0
d
V2
=‌
ε0
2
‌Ad
(‌
V
d
)
2

U=‌
ε0
2
Ad2
(‌ putting ‌‌
V
d
=E
)

A ×d= Volume of space between plates
So, energy stored per unit volume
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