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A review of basic principles of electricity & physics lec (2)

Kirchhoff’s Laws

The path by which charge may flow between the positive and negative ends of a source
of Voltage or charge is known as a circuit. Sources of current, Voltage, and resistance to
current flow are presented in a set of standard symbols that are connected together in
circuit diagrams. In such diagrams, lines represent perfect, zero resistance, conductors.
Figure A1.01 shows these standard pictures. By convention, current is said to move from
positive to negative potentials and would imply the motion of positive charges (in
actuality, most currents in man made devices result from the motion of negative
charge). Circuit diagrams frequently use arrows to indicate the direction of current flow.
In the figure, note that the current out from the top, or positive end, of the Voltage
source is identical to the current in to the bottom. The same is true of the resistor in the
circuit and indeed in any element of a circuit: The current flow in is identical to the
current flow out. This general behavior of nodes in a circuit is known as Kirchhoff’s
current law (KCL).

Kirchhoff’s current law has an analog (referred to as a “dual”) in describing Voltage
differences in circuits. Kirchhoff’s Voltage law (KVL) states that the sum of the Voltage
differences around any closed loop in a circuit must be equal to zero. In applying KVL
you must pay attention to the direction you travel in the loop, as discussed below.
Series and Parallel Connections
When electrical elements are attached end-to-end they are connected “in series”, when
the current inputs and outputs of multiple elements are held in common, the devices are
connected in parallel. Figure A1.02 shows resistors connected to a Voltage source in
series and in parallel. In the series circuit, KCL tells us that the current through all three
devices is identical, while KVL informs us that the Voltages V1 and V2 have a sum equal
to Vs. Applying KCL to the parallel circuit we see that at the two nodes, i1 + i2 = i. KVL
tells us that traveling starting at the top and going clockwise around the loop that
includes R1 and R2, V2 – V1 = 0 and that V2 = V1. Likewise, in the loop made up of Vs
and V2, V2 – Vs = 0, implying that V2 = Vs. The two Kirchhoff’s laws, in combination, are
extremely powerful organizing principles. Understanding them makes it possible to
model the behavior of the overwhelming majority of electrical circuits and devices.

Clearly, when the resistors are in series the same current flows through each. By Ohm’s
law, the voltage across R1 is equal to iR1, and the Voltage that appears across R2 is iR2.
The Voltage that appears across the Voltage source is therefore the sum of the Voltages
across each resistor, or i(R1 + R2) (Note that this is the same as the expression of KVL
for this circuit: Vs = V1 + V2. The total resistance experienced by the Voltage source in
the series circuit is thus the sum of the individual resistors: resistance in series is the
sum of the individual resistances.
In the parallel circuit, there are two paths for current to go through the resistors. Not
surprisingly more current flows through the lower resistance path, and we can compute
the individual currents by Ohm’s law. Because the Voltage (V1 and V2) across these
resistors is identical (they are connected by perfect conductors), the current, i1 in R1 is
V/R1 and i2 is V/R2. By KCL, the total current that flows from the Voltage source is
equal to the sum of the currents through the two resistors:.

which is always less than either of the resistors alone.

A review of basic principles of electricity & physics lec (1)

Introduction

While this book is not intended as a comprehensive course in electricity and magnetism

there are a few principles that are so ubiquitous in functional magnetic resonance that

they appear repeatedly throughout this text and therefore warrant this brief overview.

Charge

Electrical charge is considered to be a fundamental property of materials. Physicists

recognize that charge exists in only two forms, positive and negative, and that it is

quantal in nature, with the smallest amount of charge being that of a single electron or

proton, each being exactly 1 unit of negative or positive charge, respectively. A single

unit of charge is extremely small, of course, and charge is more commonly measured in

units of Coulombs, equivalent to about 6.242 X 1018 unit charges. Positive and negative

charges exhibit a strong attractive force, whose magnitude is proportional inversely to

the square root of the distance that separates them. In its most stable state, bulk matter

has a net charge of zero, meaning that it contains an identical number of positive and

negative charges.

Voltage

When charges become separated by distance, the presence of an attractive force between

implies an increase in potential energy, which is released when the charges are moved

together. This energy difference is known as Voltage and is measured, naturally, in

Volts. Because the potential energy of the Voltage is also measure of the force that would

tend to move the charge, it is known also as the potential difference, or simply the

potential, the “electromotive force” or the e.m.f. and these terms are used

interchangeably, which can at times be confusing. Batteries are familiar voltage sources

that rely on chemical means to store potential energy. For convenience, the units of

Volts are defined in terms of other fundamental physical constants and units. One

Joule of work is required to move one Coulomb of charge through a potential difference

of 1 Volt. In practice, this means that a Coulomb is actually defined to set unit values of

Volts and Joules. Voltage must always refer to the energy difference between two points.

It is never actually correct to discuss the Voltage at a point, though you will often see

such a statement. In those cases, the reference point is assumed implicitly, usually to

refer to a “ground” or common point in an electrical circuit.

At the atomic level charges may become separated. In some molecules, such as salts like

sodium chloride, the electronegativity of one atom (chloride) is so much greater than

that of the other (sodium) that in a covalent atomic bond between these elements the

electron or electrons are almost completely transferred from one atom to the other.

Such bonds are dissociated easily in aqueous solution so that the individual atoms now

become “ions” or charged particles. In water, the atoms of salts appear in ionic form, so

that atoms of sodium, potassium, chloride, magnesium and many others move relatively

freely of their oppositely charged complement. Not only atoms, but also molecules, can

exist in ionic form, and many proteins, for example, carry a net negative charge. Of

course some ions may be quite large and there may be physical impediments to their

motion that result in different bulk properties for ions and small charges, such as

electrons. These effects are significant in some circumstances, but in most of the

discussion that follows, and throughout most of this book, we can consider the
properties of ions interchangeably with the properties of charge.
Current and Resistance
The motion of charge is known as current; specifically, the current, i, is equal to the
change in charge, Q, with time, so that:

Where V is the Voltage, i is the current, and R the resistance. Materials whose resistance

is extremely high are termed insulators and those whose resistance is low are called,

conductors. Good insulators may have resistance of gigaOhms (109 Ohms) or more,

whereas good conductors, such as copper wire, will have resistance of microOhms. More

accurately, we refer to resistivity, which is the measured resistance normalized by the

area and length of a conductor, so that it is a material property. Most biological

materials fall in a more intermediate range with resistances of thousands to millions of

Ohms. In a perfect conductor, where the resistance is zero, the voltage at all points along

the conductor is identical. In general, moving charge from a source of higher potential

energy to lower (current flowing from positive to negative ends of a source) must result

in energy dissipation. Resistors dissipate this energy as heat.

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