CLASS 12 ELECTROSTATIC POTENTIAL AND CAPACITANCE
ELECTROSTATIC POTENTIAL AND CAPACITANCE
✤ Potential difference: between 2 points in an electric field may be defined as the amount of work done in moving a unit pot charge from 1 point to the
other against the electrostatics force.
V=VB-VA = WAB /qo(Volt)
[ML-2T-3A-1]
1. It is a scalar quantity. It may be +ve and -ve
Electric Potential at a point in an electric field is the amount of work done in moving a unit positive charge from infinity to that point against the electrostatics force.
V = work done / Charge (Volt)
work done to maintain the flow of electrons in a circuit is called Potential.
Electric potential due to a point charge
V= 1/4πεr . q/r
Notes :-
VA = 1/4πE (q1/a + q2/b + q3/c)
VB = 1/4πE (q1/b + q2/b + q3/c)
Vc = 1/4πE (q1/c + q2/c + q3/c)
Electric Potential due to a Dipole
* At axial point: V = 1/4πε₀ * P/(r²-a²)
* At equatorial point: V = 0
* At any general point:
V = 1/4πε₀ * Pcosθ/(r²-a²cosθ)
Special Case
* When the point lies on the axial line of the dipole θ=0° or 180°, and
V = ± 1/4πε₀ * P/(r²-a²)
* The potential has greatest positive on the greatest negative value
* When the point lies on the equatorial lines of the dipole θ=90°, and V=0
* The electric field at such points is non-zero.
Relation btw 'E' & 'V':
E = [-dv/dr] - potential gradient
Its Shows direction of the potential
* If E=0 then V=constant
* dv/dr is a vector quantity.
✤ Equipotential Surface :- Any surface that has same electric potential at every point on it is called an equipotential surface
1) VB=VC 2) WAB = WAC
✤ Properties of Equipotential Surface
i) No-work is done in moving a test charge over an equipotential surface.
ii) Electric field is always normal to the equipotential surface at every point.
iii) Equipotential surfaces are closer together in the regions of strong field and farther apart in the regions of weak field.
iv) No 2 equipotential surface can intersect each other.
✤ Equipotential Surfaces of Various charge systems.
i) a point charge
ii) a uniform electric field
iii) 2 point +q & -q separated by a small distance.
iv) 2 point to +q separated by a small distance.
✤ Electric Potential Energy: of a system of point charges may be defined as the amount of work done in assembling the charges at their locations by bringing them in from infinity.
- Scalar quantity
Multiples and submultiples of eV
1meV (milli electron volt) = 10-3 eV = 1.6x10-22 J
1keV (kilo electron volt) = 103 eV = 1.6x10-16 J
1 MeV (million electron volt) = 106 eV = 1.6x10-13 J
1GeV (giga electron volt) = 109 eV = 1.6x10-10 J
1Tev (tera electron volt) = 1012 eV = 1.6x10-7 J
Potential Energy of a Dipole in a uniform Electric field
U=pε(cosθ₁ -cosθ₂)
Position of stable equilibrium θ = 0°
U = -pεcos0°= -pε (when P is || to External Field minimum)
Position of zero energy θ= 90°
U=-pεcos90° = 0 (⊥ to external field)
Position of unstable equilibrium θ=180°
U=-pεcos180° = +pε (max when P is anti-parallel to External field)
W₂=-u
Behaviors of Conductors in Electrostatics Fields
i) Net to electrostatic field is zero in the interior of a conductor.
ii) Just outside the surface of a charged conductor electric field is normal to the surface.
iii) The net charge in the interior of a conductor is zero & any excess charge resides at its surface.
iv) Potential is constant within and on the surface of conductor
v) Electric field at the surface of a charged conductor is proportional to the surface charge density. E=σ/ε₀
vi) Electric field is zero in the cavity of a hollow charged conductor.
Electrical Capacitance of a conductor
The electrical Capacitance of a conductor is the measure of its ability to hold electric charge.
Capacitance of a conductor may be defined as the charge required to increase the potential of the conductor by unit amount.
Depends upon the following factors:-
i) Size & shape of the conductor.
ii) Nature (permittivity) of the surrounding medium.
iii) Presence of the other conductors in its neighbourhood.
Unit -> Farad, C/V, C²/J
1 millifarad = 1mF = 10⁻³F
1 microfarad = 1µF = 10⁻⁶F
1 picofarad = 1pF = 10⁻¹²F
Dimensions = [M⁻¹ L⁻² T⁴ A²]
Capacitance of an Isolated Spherical Capacitor.
C=4πϵ₀R
Principle of Conductor
Derive the expression for the capacitance of a parallel plate capacitor whose plates are separated by a dielectric medium. Whenever 2 neutral conductors are placed near by.
q=CV
§ C' does not depend upon q and v.
Capacitor
Is an arrangement of 2 conductors separated by an insulating medium that is used to store electric charge & electric energy.
Here E=σ/ϵ₀
V= EXd
P.d. blw the plates.
E=σ/2ϵ₀ -> at ∞ plane sheet
Parallel Plate Capacitor C=ϵ₀A/d
Depends on factors:
i) Area of the plates (∝A)
ii) Distance b/w the plates (∝1/d)
iii) Permittivity of the medium bhw the plates (∝ϵ₀)
iv) If a dielectric 'k' is filled btw the plates, then.
C'=Kϵ₀A/d
or C'=KCo
• Combination of capacitors in Series
added end to end connected with any source.
1/c = 1/c₁ + 1/c₂ + 1/c₃
- equivalent capacitance is < small individual capacitance.
- Charges on each capacitor is same
- P.D across any capacitor is α-1/c
• Combination of capacitors in Parallel
If first plate of all the capacitors is connected to a point & another plate of all the capacitors are connected to another point.
Cp=c₁ + c₂ + ... + cn
- equivalent capacitance is > largest individual capacitance.
- Charges on each capacitor α-c
- P.D across any capacitor is same.
• Energy stored in a capacitor
Work done in charging the capacitor is stored as its electrical potential energy.
U=1/2 ⋅ Q²/C = 1/2 ⋅ C V²=1/2 ⋅ Q V Joules
o Capacitance of the capacitor if dielectric slab is inserted btw the plates:
C = E₀A/d-t+(t/E)
1. P.D btw the plates
(i) V = E(d-t) + E․t (ii) E = V/d
Use of Capacitors
i) To produce electric field of desired patterns.
eg:-Millikan's experiment.
ii) In radio circuits for turning.
iii) Tank circuit of oscillators.
iv) Producing rotating magnetic fields in induction motors.
v) Powers supplies for smoothing the rectified current.
Effect of dielectric on various parameters
Battery disconnected from the capacitor Battery kept connected across the capacitor
Charge Q=Qo (constant) Q=kQo
P.D V= Vo/k V=Vo (constant)
E.F E= Eo/k E=Eo (constant)
Capacitance C= kCo C=kCo
Energy stored U= Uo/k U=k/Uo
Note :
C1 = EoAK1 /2d C2 = EoAK2/2d
and Total capacitance
C = C1+C2
= EoA/d([K1+K2]/2)
1. Suppose we have n drops each having radius r, charge q, Potential v, energy u and capacitance c combine each other then →
1) charge on bigger drop: Q = nq
2) Radius of bigger drop: R = (n)⅓ r
3) Potential of bigger drop: V' = (n)⅔V
4) Energy of big drop: U' = (n)⁵/³U
5) surface charge density = (n)⅔ σ.
6) Capacitance c' = (n)⅓C.
✤ Common potential: when two different capacitors connect by a wire then charge flows from high potential to low potential till potential of two capacitor become equal.
V = C₁V₁+C₂V₂ / C₁ + C₂
Loss of energy: ΔU = C₁C₂ (V₁-V₂)² / 2(C₁+C₂)
✤ Energy density: Energy stored in a capacitor per unit volume.
Energy density = Energy volume
= ½ ε₀E² J/m³
Note: If n identical plates are connected as shown electric field they form (n-1) capacitors then plate
Note: 1) Potential energy spend by battery in charging = qV, whereas energy stored in capacitor = ½ qV
2) Rest energy is lost in charging the capacitor.
Note: When charge is shared b/w two bodies, their potential becomes equal.
The charge acquired are in the ratio of their capacities. No charge is lost, but some energy is lost.
