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CBSE Class 12 Physics 2020 Delhi Set 1 Paper

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Question : 21 of 37
Marks: +1, -0
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 ABAB consisting of charge +q+q and q-q and of length 2a2a is placed in uniform electric field EE making an angle θ\theta with the direction of electric field.
Force acting on q-q is qE-q E
Force acting on +q+q is qEq E
These two forces are equal and opposite to each other. Hence, a torque on the dipole is developed.
Torque == Force ×\times perpendicular distance between the forces
  Or,    τ=qE×2asinθ\;\text{Or,}\;\; \tau = q E \times 2a \sin \theta
  Or,    τ=(q×2a)Esinθ\;\text{Or,}\;\; \tau = (q \times 2a) E \sin \theta
  τ=  PE  sinθ\therefore \;\tau = \;\text{PE}\; \sin \theta
    τ=PEsinθ      (where  P  is dipole moment)  \therefore \;\; \tau = PE \sin \theta \;\; \;\text{(where}\; P \;\text{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 VdQV dQ, where VV is the voltage across the capacitor.
dU=VdQ=QCdQdU = VdQ = \frac{Q}{C} dQ
\therefore Energy stored ==
U=VdQ=1CQdQU = \int VdQ = \frac{1}{C} \int QdQ =12Q2C=12CV2= \frac{1}{2} \frac{Q^2}{C} = \frac{1}{2} CV^2 ......(i)
Energy density is defined as the total energy per unit volume of the capacitor.
For a parallel plate capacitor,
C=Aε0dC = \frac{A \varepsilon_0}{d}
Putting in eqn (i)
U=12Aε0dV2U = \frac{1}{2} \frac{A \varepsilon_0}{d} V^2 =ε02Ad(Vd)2= \frac{\varepsilon_0}{2} Ad \left( \frac{V}{d} \right)^2
U=ε02Ad2(  putting    Vd=E)U = \frac{\varepsilon_0}{2} Ad^2 \left( \;\text{putting}\; \; \frac{V}{d} = E \right)
A ×d=\times d = Volume of space between plates
So, energy stored per unit volume
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