24Capacitance_notes

A series connection
The equivalent capacitor
• A potential dierence V is applied between points a and b • A +ve charge Q builds up on the top plate of C1 • This attracts an equal amount of −ve charge to the bottom plate of C1 • These −ve charges had to come from the top plate of C2 , which acquires +ve charge Q • This attracts an equal amount of −ve charge to the bottom plate of C2 • From Eq. (24.1), V1 = • Adding potential dierences: V = V1 + V 2 = Q • Equivalent capacitance is Ceq = Q/V , thus 1 V 1 1 = = + Ceq Q C1 C2 Q , C1 Q C2
capacitance is the measure of the ability of a capacitor to store electric potential energy
• We will see below that the capacitance of a capacitor depends only on the shapes and sizes of the
• The above expression for E is the same as that for a point charge, so the potential is also the same as that for a point charge, i.e., V = Q/4π 0 r • Thus, Vab = • The capacitance is therefore C= Q = 4π Vab
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24.1 Capacitors and Capacitance
Denition of a capacitor with charge Q
conductors, and on the nature of the insulating material between them
Parallel-plate capacitor
• The simplest form of capacitor consists of two parallel conducting plates, each with area A, and separated by a distance d • When the separation of the plates is small compared to their size, the electric eld between the plates
0
Q 4π 0
1 1 − ra rb
=
Q r b − ra 4π 0 r a r b
ra rb rb − ra
Example 24.4: Capacitance of a cylindrical capacitor
• By Gauss's law, the charge on the outer cylinder does not contribute to the eld between the cylinders • From Ex. 23.10, the potential outside a charged cylinder a distance r from the axis is V = • Thus, Vab = λ 2π 0 r0 λ ln 2π 0 r = λ rb ln 2π 0 ra
potential dierence Vab between the conductors equal to the voltage of the battery
1
Denition of capacitance C
• The electric eld at any point in the region between the conductors is proportional to Q • Thus, the potential dierence Vab between the conductors is also proportional to Q • Their ratio is called the
3
Test your understanding
1. If you double the amount of charge on a capacitor, how does the capacitance change?
24.2 Capacitors in Series and Parallel
Capacitors in series
• By Gauss's law, the charge on the outer shell does not contribute to the eld between the shells • From Ex. 22.5, the electric eld outside a charged shell a distance r from the centre is E= Q 4π 0 r2
V2 =
1 1 + C1 C2
Capacitors in series: Summary
• In a series connection, all the capacitors have the same charge • The reciprocal of the equivalent capacitance equals the sum of the reciprocals of the individual capac-
itances:
1 1 1 1 = + + + ··· Ceq C1 C2 C3
(24.5)
4
Capacitors in parallel
A parallel connection to it by conducting wires
The equivalent capacitor
• The upper plates of the two capacitors are at the same potential as point a, since they are connected • Similarly, the lower plates of the two capacitors are at the same potential as point b • Thus, the potential dierence for the two capacitors is equal to V • From Eq. (24.1), Q1 = C1 V , • The Q2 = C2 V
is essentially uniform
Capacitance of a parallel-plate capacitor (in vacuum)
• Surface charge density on each plate is σ = Q/A • From Ex. 22.8, the electric eld magnitude between the plates is E= σ
capacitance C of the capacitor:
C= Q Vab
(24.1)
• Unit: coulomb per volt (C/V) ≡ farad (F)
Signicance of capacitance
• The greater the capacitance C of a capacitor, the greater the charge Q it can acquire for a given potential dierence Vab • The greater the charge Q of the capacitor, the greater the amount of stored electric potential energy • Thus,
ln
r0 r0 − ln ra rb
• The total charge Q in a length L is Q = λL, so the capacitance of this length is C= • The capacitance per unit length is therefore C 2π 0 = L ln(rb /ra ) Q 2π 0 L = Vab ln(rb /ra )
0
=
Q 0A Qd 0A A d
• The potential dierence between the two plates is Vab = Ed = • Thus, C= Q = Vab
0
(24.2)
2
Example 24.3: Capacitance of a spherical capacitor
total charge Q of the two capacitors is
Q = Q1 + Q2 = (C1 + C2 )V
• Equivalent capacitance Ceq is Ceq =
Q = C1 + C2 V
Capacitors in parallel: Summary
• In a parallel connection, the potential dierence for all individual capacitors is the same • The equivalent capacitance equals the sum of the individual capacitances: Ceq = C1 + C2 + C3 + · · ·
ຫໍສະໝຸດ Baidu
Charging a capacitor
• Assume capacitor has zero charge initially • Connect the conductors to opposite terminals of a battery • Electrons are transferred from one conductor to the other • The charge of the capacitor increases until it reaches some maximum value Q • This gives a xed
• Any two conductors separated by an insulator (or a vacuum) form a • Such a capacitor is said to have charge Q
capacitor
• Usually, the two conductors have charges with equal magnitude and opposite sign, say +Q and −Q
24 Capacitance and Dielectrics Edward Teo
Introduction
• A capacitor is a simple device that stores electric charge, or electric potential energy • Capacitors have a tremendous number of practical applications: e.g., camera ashes, debrillators,