**Q3). A 3 mF capacitor and a 6 mF capacitor are connected in series across an 18 V battery. Determine the equivalent capacitance and total charge deposited.Given:

V = 18 VC1= 3 mFC2= 6 mF

Find:

Ceq=?Q=?First determine equivalent capacitance of C1 and C2:Next, determine the charge

1 / Ceq = 1 / C1 + 1 / C2 11Two capacitors in series and the equivalent capacitor. V = V1 + V2 and Q = C V

*

Capacitors in parallelAdding capacitors in parallel, Vtot = V1 = V2, so

** Parallel combinationConnecting a battery to the parallel combination of capacitors is equivalent to introducing the same potential difference for both capacitors, A total charge transferred to the system from the battery is the sum of charges of the two capacitors,By definition,

Thus, Ceq would be

Ceq = C1 + C2 Two capacitors in parallel and the equivalent capacitor. Q = Q1 + Q2 and Q = C V

**AA+Q-Qd The parallel-plate capacitorThe capacitance of a device depends on the geometric arrangement of the conductors

where A is the area of one of the plates, d is the separation, e0 is a constant (permittivity of free space),

e0= 8.8510-12 C2/Nm2

**Q 4). A parallel plate capacitor has plates 2.00 m2 in area, separated by a distance of 5.00 mm. A potential difference of 10,000 V is applied across the capacitor. Determine the capacitance the charge on each plateGiven:

DV=10,000 VA = 2.00 m2d = 5.00 mm

Find:C?Q=?

Solution:Since we are dealing with the parallel-plate capacitor, the capacitance can be found asOnce the capacitance is known, the charge can be found from the definition of a capacitance via charge and potential difference:

** Energy stored in a charged capacitorConsider a battery connected to a capacitorA battery must do work to move electrons from one plate to the other. The work done to move a small charge q across a voltage V is W = V q. As the charge increases, V increases so the work to bring q increases. Using calculus we find that the energy (U) stored on a capacitor is given by:VVqQ

Energy in capacitorsW = qVave

*Find electric field energy density (energy per unit volume) in a parallel-plate capacitorExample: electric field energy in parallel-plate capacitorRecall

Thus,

and o, the energy density is

Example 1 : Parallel-Plate CapacitorCalculate field strength E as a function of charge Q on the platesIntegrate field to calculate potential V between the platesQ=CV, C = Q/V

Example 1: Parallel plate capacitorWhen the plates are charged, one plate has charge +Q, and the other -Q, so we say that the charge on the capacitor is Q.For parallel plates, V = Ed and E = 0, where = Q/A is the charge density on the plates, so

Example 2: A charged sphere of radius RIf the charge on the sphere is Q, the potential at the surface is V = kQ/RThe capacitance is then

Charging capacitors in RC circuitsWhen switch is first closed, uncharged capacitor acts like a wire, with no voltage drop across it (t = 0)After a long time, once the capacitor is fully charged, it acts like an open switch (t infinity)When switch is first closed on a charged capacitor, it acts like a battery

CapacitorsA capacitor is a passive element that stores energy in its electric field. A capacitor consists of two conducting plates separated by an insulator (or dielectric). When a voltage source is connected to the capacitor, the source deposits a positive charge, +q, on one plate and a negative charge, q, on the other. The amount of charge is directly proportional to the voltage so that +q-q+q-q

v

+

-

C

+

-

C

v

CapacitorsC, called the capacitance of the capacitor, is the constant of proportionality. C is measured in Farads (F). From

we define:Capacitance is the ratio of the charge on one plate of a capacitor to the voltage difference between the two plates, measured in Farad (F). Thus, 1F = 1 coulomb/volt

In reality, the value of C depends on the surface area of the plates, the spacing between the plates, and the permitivity of the material.

The capacitor has the following important properties:

When the voltage across a capacitor is constant (not changing with time) the current through the capacitor:

i = C dv/dt = 0

Thus, a capacitor is an open circuit to dc. If, however, a dc voltage is suddenly connected across a capacitor, the capacitor begins to charge (store energy).

2.The voltage across a capacitor must be continuous, since a jump (a discontinuity) change in the voltage would require an infinite current, which is physically impossible. Thus, a capacitor resists an abrupt change in the voltage across it, and the voltage across a capacitor cannot change instantaneously, whereas, the current can.

The capacitor has the following important properties:3.The ideal capacitor does not dissipate energy. It takes power from the circuit when storing energy and returns previously stored energy when delivering power to the circuit.

4.A real, non-ideal, capacitor has a leakage resistance which is modeled as shown below. The leakage resistance may be as high as 100M, and can be neglected for most practical applications.

In this course we will always assume that the capacitors are ideal.

C

S

R

*Energy Storing in Capacitor+-viC

*

The charge on a capacitor is an integration of current through the capacitor. Hence, the memory effect counts. +-viC

Energy Density

Energy stored in the capacitorThe instantaneous power delivered to the capacitor is The energy stored in the capacitor is thus

**In the circuit shown V = 48V, C1 = 9mF, C2 = 4mF and C3 = 8mF.determine the equivalent capacitance of the circuit,(b) determine the energy stored in the combination by calculating the energy stored in the equivalent capacitance.Example: stored energy

**In the circuit shown V = 48V, C1 = 9mF, C2 = 4mF and C3 = 8mF.(a) determine the equivalent capacitance of the circuit,(b) determine the energy stored in the combination by calculating the energy stored in the equivalent capacitance,

First determine equivalent capacitance of C2 and C3:The energy stored in the capacitor C123 is thenNext, determine equivalent capacitance of the circuit by noting that C1 and C23 are connected in seriesGivenV = 48 C1= 9 mFC2= 4 mFC3= 8 mFFind:Ceq=?U =?

*Example Find the equivalent capacitance seen between terminals a and b of the circuit

*Example Solution:

InductorsEnergy Storage Devices

*Energy Storage FormAn inductor is a passive element designed to store energy in the magnetic field while a capacitor stores energy in the electric field.

* InductorsAn inductor is made of a coil of conducting wire

*****Here is a setup that is simple enough for a direct calculation:-Gauss Law to calculate the E-field between the plates as a function of charge Q Integrate electric field to get the potential (voltage, V) between the platesQ=CV = whatever is the constant of proportionality between Q and V muist be C!Note : with a charge +Q on a plate area A, the charge density s=Q/A Coulombs/M2