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Revision Notes on Electrostatic Potential and Capacitance:-
(a) Electric potential, at any point, is defined as the negative line integral of electric field from infinity to that point along any path.
(b) V(r) = kq/r
(c) Potential difference, between any two points, in an electric field is defined as the work done in taking a unit positive charge from one point to the other against the electric field.
W_{AB} = q [V_{A}-V_{B}]
So, V = [V_{A}-V_{B}] = W/q
Units:- volt (S.I), stat-volt (C.G.S)
Dimension:- [V] = [ML^{2}T^{-3}A^{-1}]
Relation between volt and stat-volt:- 1 volt = (1/300) stat-volt
E = -dV/dx = --dV/dr
V = (1/4π ε_{0}) (q/r)
V = (1/4π ε_{0}) [q_{1}/r_{1} + q_{2}/r_{2} + q_{3}/r_{3}]
= V_{1}+V_{2}+ V_{2}+….
(a) Outside, V_{out} = (1/4π ε_{0}) (q/r)
(b) Inside, V_{in} = - (1/4π ε_{0}) (q/R)
(c) On the surface, V_{surface} = (1/4π ε_{0}) (q/R)
(b) Inside, V_{in} = (1/4π ε_{0}) [q(3R^{2}-r^{2})/2R^{3}]
(d) In center, V_{center} = (3/2) [(1/4π ε_{0}) (q/R)] = 3/2 [V_{surface}]
(a) Common potential, V = (1/4π ε_{0}) [(Q_{1}+Q_{2})/(r_{1}+r_{2})]
(b) q_{1} = r_{1}(Q_{1}+Q_{2})/(r_{1}+r_{2}) = r_{1}Q/ r_{1}+r_{2} ; q_{2} = r_{2}Q/ r_{1}+r_{2}
(c) q_{1}/q_{2} = r_{1}/r_{2} or σ_{1}/ σ_{2} = r_{1}/r_{2}
V (r,θ) = qa cosθ/4πε_{0}r^{2} = p cosθ/4πε_{0}r^{2}
(a) Point lying on the axial line:- V = p/4πε_{0}r^{2}
(b) Point situated on equatorial lines:- V = 0
(a) R = n^{1/3}r
(b) Q = nq
(c) V = n^{2/3}V_{small}
(d) σ = n^{1/3} σ_{small}
(e) E = n^{1/3} E_{small}
W = U = (1/4πε_{0}) (q_{1}q_{2}/r_{12}) = q_{1}V_{1}
W = U = (1/4πε_{0}) (q_{1}q_{2}/r_{12} + q_{1}q_{3}/r_{13} + q_{2}q_{3}/r_{23})
(a) If θ = 90º, then W = 0
(b) If θ = 0º, then W = -pE
(c) If θ = 180º, then W = pE
K. E = ½ mv^{2 }= eV
Conductors:- Conductors are those substance through which electric charge easily.
Insulators:- Insulators (also called dielectrics) are those substances through which electric charge cannot pass easily.
Capacity:- The capacity of a conductor is defined as the ratio between the charge of the conductor to its potential
C = Q/V
Units:-
S.I – farad (coulomb/volt)
C.G.S – stat farad (stat-coulomb/stat-volt)
Dimension of C:- [M^{-1}L^{-2}T^{4}A^{2}]
C = 4πε_{0}r
Capacitor:- A capacitor or a condenser is an arrangement which provides a larger capacity in a smaller space.
Capacity of a parallel plate capacitor:-
C_{air }= ε_{0}A/d
C_{med} = Kε_{0}A/d
Here, A is the common area of the two plates and d is the distance between the plates.
C = ε_{0}A/[d-t+(t/K)]
Here d is the separation between the plates, t is the thickness of the dielectric slab A is the area and K is the dielectric constant of the material of the slab.
If the space is completely filled with dielectric medium (t=d), then,
C = ε_{0}KA/ d
(a) C_{air} = 4πε_{0}R
(b) C_{med} = K (4πε_{0}R)
(a) When outer sphere is earthed:-
C_{air} = 4πε_{0} [ab/(b-a)]
C_{med} = 4πε_{0} [Kab/(b-a)]
(b) When the inner sphere is earthed:-
C_{1}= 4πε_{0} [ab/(b-a)]
C_{2} = 4πε_{0}b?
Net Capacity, C '=4πε_{0}[b^{2}/b-a]
Increase in capacity, ΔC = 4π ε_{0}b
It signifies, by connecting the inner sphere to earth and charging the outer one we get an additional capacity equal to the capacity of outer sphere.
C_{air} = λl / [(λ/2π ε_{0}) (log_{e }b/a)] = [2π ε_{0}l /(log_{e }b/a) ]
C_{med} = [2πKε_{0}l /(log_{e }b/a) ]
W = ½ QV = ½ Q^{2}/C = ½ CV^{2}
U = ½ ε_{0}E^{2} = ½ (σ^{2}/ ε_{0})
This signifies the energy density of a capacitor is independent of the area of plates of distance between them so long the value of E does not change.
?(a)
(i) Capacitors in parallel:- C = C_{1}+C_{2}+C_{3}+…..+C_{n}
_{ ?}
The resultant capacity of a number of capacitors, connected in parallel, is equal to the sum of their individual capacities.
(ii)V_{1}= V_{2}= V_{3} = V
(iii) q_{1} =C_{1}V, q_{2} = C_{2}V, q_{3} = C_{3}V
(iv) Energy Stored, U = U_{1}+U_{2}+U_{3}
(b)
(i) Capacitors in Series:- 1/C = 1/C_{1 }+ 1/ C_{2 }+……+ 1/C_{n}
?
The reciprocal of the resultant capacity of a number of capacitors, connected in series, is equal to the sum of the reciprocals of their individual capacities.
(ii) q_{1} = q_{2} = q_{3} = q
(iii) V_{1}= q/C_{1}, V_{2}= q/C_{2}, V_{3}= q/C_{3}
(a) Energy stored in a series combination of capacitors:-
W = ½ (q^{2}/C_{1}) + ½ (q^{2}/C_{2}) + ½ (q^{2}/C_{3}) = W_{1}+W_{2}+W_{3}
Thus, net energy stored in the combination is equal to the sum of the energies stored in the component capacitors.
(b) Energy stored in a parallel combination of capacitors:-
W = ½ C_{1}V ^{2} +½ C_{2}V ^{2} + ½ C_{3}V ^{2} = W_{1}+W_{2}+W_{3}
The net energy stored in the combination is equal to sum of energies stored in the component capacitors.
(a) F = ½ ε_{0}E^{2}A
(b) F = σ^{2}A/2ε_{0}
(c) F=Q^{2}/2ε_{0}A
F = (Q^{2}/2C^{2}) (dC/dx) = ½ V^{2} (dC/dx)
V = [C_{1}V_{1}+ C_{2}V_{2}] / [C_{1}+C_{2}] = [Q_{1}+Q_{2}]/ [C_{1}+C_{2}]
ΔQ = [C_{1}C_{2}/C_{1}+C_{2}] [V_{1}-V_{2}]
ΔU = ½ [C_{1}C_{2}/C_{1}+C_{2}] [V_{1}-V_{2}] ^{2}
(a) Q = Q_{0}(1-e^{-t/RC})
(b) V = V_{0}(1-e^{-t/RC})
(c) I = I_{0}(1-e^{-t/RC})
(d) I_{0} = V_{0}/R
(a) Q = Q_{0}(e^{-t/RC})
(b) V = V_{0}(e^{-t/RC})
(c) I = I_{0}(e^{-t/RC})
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