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# Electrical Current Study Guide (page 2)

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Updated on Sep 27, 2011

#### Solution

First, convert all quantities to SI and then set your formulas to determine I.

The electron charge is:

qe = –1.6 · 10–19 C

And because we have a number of 108 electrons, the total charge is:

Δq = 108 · qe = 108 · (–1.6 · 10–19C) = –1.6 · 10–11C

Δt = 1 ms = 10–3s

I = ?

By the definition of the electrical current:

Because the current is created by electrons moving through the conductor, and considering the previous note, the current is taken to be opposite to the velocity v.

## Electrical Resistance and Ohm's Law

Mechanical motion or heat transfer are sometimes associated with loss of energy (as, for example, with friction). The same situation is encountered in electricity. The same battery hooked up to different wires will establish currents of different values. Therefore, we can introduce the concept of resistance: electrical resistance.

Electrical resistances are measured in volts per ampere, and the unit is called an ohm (Ω) after Georg Ohm (1789–1854).

The electrical resistance varies with some factors, but for most materials, the ratio stays the same for a large range of currents and voltages. In these cases, the resistance is a constant, and this observation is known as Ohm's law:

= constant

Electrical resistance R varies with physical properties of the wires (length and cross-sectional area) and with the nature of the material (coefficient known as resistivity, p). The value of the resistivity helps classify materials into conductors, semiconductors, and insulators. Electrical conductors have small resistivities and conduct electrical current well (examples include: copper and silver), whereas insulators hinder the flow of charges (examples include: wood and Teflon). Semiconductors have a varying resistivity, and in certain conditions, they can behave as conductors and in other conditions as insulators. These are intrinsic properties of the materials and are highly dependent on the type of bonding between subatomic particles.

With this new insight, we can also define the electrical resistance in the following way. Electrical resistance R is proportional to the electrical resistivity, p, of the material and the length and it is inversely proportional to the cross-sectional area A.

### Electrical Resistance

Electrical resistance is the ratio of the voltage v applied across the ends of a material to the current I established through the material.

#### Example

Determine the unit for the electrical resistivity.

#### Solution

We start with the definition and solve for resistivity by multiplying each side with A/L:

So, the emerging unit is

A few examples of the most encountered conductor materials and their resistivity are listed in Table 14.1.

Most diagrams containing a resistor are going to represent it in a manner similar to that shown in Figure 14.3.

### Electrical Resistance

Electrical resistance R is proportional to the electrical resistivity, p, of the material and the length and is inversely proportional to the cross-sectional area A.

## Electrical Capacitance

Wires are not the only electrical devices that perform a function when a potential difference is applied. Another important category of devices is capacitors.

Capacitors are systems composed of two conductors placed in proximity but not in contact. The space between the conductors is filled with insulating materials called dielectrics. The plates of the capacitor are charged with equal and opposite charges. An electrical field is established as shown in Figure 14.4.

The amount of charge that a capacitor can store on each plate is proportional to the voltage V applied and the capacitance C.

q = C· V

If we solve for the capacitance:

The capacitance is measured in SI, in coulombs per volt in a unit called a farad (F), named after Michael Faraday (1791–1867). Usual capacitances are in the range of microfaradds to picofarads.

Dielectrics placed in between the plates increase the capacitance and therefore the charge of the capacitor. They decrease the field established between the plates. A measure of this property is the dielectric constant K that represents the ratio of the field without and with dielectric. Therefore, this coefficient is unitless.

Similar to the dependence noted previously for electrical resistance, the electrical capacitance is dependent upon intrinsic properties of the dielectric and on the geometry of the plates. For a parallel plate capacitor, such as the one shown previously in Figure 14.4, the dependence is summarized in the expression:

The other constant is called permittivity of vacuum and is expressed by:

ε0 = 8.8 · 10–12C2/(N · m2)

### Electrical Capacitance

The electrical capacitance is proportional to the dielectric constant · K and the area of the plates A and inversely proportional to the distance between the plates d.

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