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Review Material

Questions

Capacitors are used to store energy (as in batteries, etc.). It takes a lot of capacitance to store a lot of charge though.

The direction relationship between capacitance to charge is shown the in equation:

where C is the capacitance, V is the voltage, and q is the charge.

[C] = Farads = F

[V] = V

[q] = C

In order to put capacitors to good use, you have to hook em up to a circuit and let charge flow through it. As in direct current circuits, there are two ways to do that:

Capacitors in Parallel

Well what does is sound like? The capacitors are hooked up parallel to each other after all =)

Let's say the charge that flows through capacitor #1 is q₁, and the charge that flows through capacitor #2 is q₂. Then the total charge flowing through the circuit is the sum of the two charges.

Think of it this way. Charge can flow through either C₁ or C₂, but it will bypass the other one.

The potential difference across the capacitors are constant (meaning the V of C₁ and the V of C₂ are the same, and V is equal to the voltage of the battery). Same reasoning: the voltage of the battery (which gives charged particles their energy to flow throughout the circuit) can go either through C₁ or C₂.

Thus, for parallel plate capacitors

Capacitors in Series

These capacitors are hooked up directly in the "flow path" of each other. The charge flowing this way will have to stop at C1 first then whatever is left over (not stored by C1) will flow on to C2.

In series, the charge of each capacitor is equal to the charge of the entire circuit.

q1 = q2 = qn

By the law of conservation of charge this is verified.

And we have that

From the diagram (since these capacitors are in series and charge q is a constant value) it may be discerned that V, it's voltage or potential difference is equal to to the sum of its parts (the V of each capacitor)

V = V1 + V2

and expressing that in terms of q and C:

since the q's are equivalent and we want to know the relationship of the capacitances, we can cancel these out (it's all in the math) to get

the sum of the inverses of each capacitance is the inverse of the total capacitance. (no you can't just say C_T = C₁ + C₂)

Stored Energy of a Capacitor

How does a capacitor work? True, if both plates are at the same potential (neutral or not charged), little work would be required to transfer charge from one plate to another. However, once that is accomplished, the plates are beginning to accumulate charge and now their potentials are not the same. Work becomes a factor in moving the charge across. The increase in work required for each increase in charge (that is being transferred) is the product of the potential difference at that point and the increase in the charge of the plate(s) caused by the transfer.

and V = q/C, so

(the area under the straight line). Numerous ways to say the same thing:

Dielectrics

Dielectrics are insulators that have been put in between the plates of a capacitor. Well what's that for? No it doesn't stop flow but it does decrease the potential difference between the plates, thus increasing the capacitance of the capacitor.

If the dielectric fills up the entire space between the conductors, the capacitance of the dielectric will increase by a factor of (i.e. mulitiplied by) a value which we label K, the dielectric constant. But K depends on the type of insulating material.

Basically, when the capacitor doesn't have a dielectric (air or vacuum), it will follow this formula:

But with a more substantial dielectric, there will be a change to the potential difference (it's being reduced for better capacitance).

Thus, by pluging in this, we get

where q₀ and V₀ is, respectively the charge and potential difference without the dielectric.

Some Dielectric Constants

Direct Current Circuits

A circuit has been built to transfer and store energy. Capacitors  have been used to cover the storage part, so now let's get down to the transfer part. Now we want to define voltage, current, and resistance, integral concepts to direct current circuits.

The flow of charge through these wires is called current. Current is, more exactly the rate at which charge flows through a point (or surface area).

Current is represented with the symbol I, and is, aptly, equal to the charge that flows through a point per second:

[I] = Ampere = A

Now, as humans, we want to be able to control things. First and foremost, our environment and energy sources. Current can be controlled through the use of resistors. Basically it is used to reduce current flow.

Resistance is the measure of the effectiveness to which a resistor reduces current. Directly proportional to the length of the resistor, l, and inversely proportional to it's cross-sectional area, A, it looks sorta like this:

The constant of proportionality is p, called the resistivity.

[R] =

Resistors' resistance may vary, but there are ways to compute resistance. But we can't use resistivity (p) because it too is dependent on two things! (A constant that isn't always constant hmmm...) Thus, there are two factors that determine how the resistance of a circuit changes:

• coefficient of resistivity,  
• temperature

Resistivity is characteristic for each kind of material and is a constant of proportionality for resistance. Resistivity and temperature are actually interrelated, thus making temperature a factor in determining resistance.

[Resistivity] = [p] = ·m

p = p0(1 + /\T) = p0[1 + (T - T0)]

similarly,

R = R0(1 + /\T)

Some Coefficients of Resistivity

Material	Coefficient of Resistivity () °C-1	Resistivity (p)
Aluminum	3.9 x 10-3	2.82 x 10-8
Carbon	-0.5 x 10-3	3.5 x105
Copper	3.9 x 10-3	1.7 x 10-8
Germanium	-48 x 10-4	0.46
Gold	3.4 x 10-3	2.44 x 10-8
Iron	5.0 x 10-3	10.0 x 10-8
Lead	3.9 x 10-3	2.2 x 10-9
Platinum	3.92 x 10-3	11 x 10-8
Silicon	-75 x 10-3	640
Silver	3.8 x 10-3	1.59 x 10-8
Tungsten	4.5 x 10-3	5.6 x 10-8

The last in our elements an important DC circuit equation is V, which stands for the voltage of the system. Basically this is otherwise known as the potential difference of the circuit (need a refresher? click here).

Well what does this all culminate to?

Nothing other than Ohm's Law.

Read up on Ohm!

Now, we can apply this to circuits.

Kirchoff's Laws

Basically these are conservation laws.

1. (What goes in, must come out) The sum of all the currents entering a junction point must be equal to the sum of all the currents leaving that junction point.

2.Sum of the voltage drops or rises across all the parts of the circuit around any (possible) closed loop must equal zero.

Energy and Power

We come across a rule that expresses P (power) in terms of current and voltage.

This, by substitution with Ohm's law, will enable us to express power in terms of R as well. But the fact that P= W/t will come back and haunt you >=)

[Back to the Top Practice Questions]

• Legend
• Basic Circuit Construction
• Capacitance
• Energy
• Dielectrics
• Direct Current Circuits
• Ohm's Law
• Kirchoff's Rules
• Energy and Power
• Practice Questions

Java Interactions and Demonstrations

• Two-Resistor Circuit
• Four-Resistor Circuit
• Kirchoff's Rules 1
• Kirchoff's Rules 2
• Kirchoff's Rules 3
• Kirchoff's Rules 4
• Kirchoff's Rules 5

Links

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• Discover Magazine
• OMNI Magazine
• Popular Science
• Scientific American

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