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Electricity is the flow of electrons through a conductive material. Metals are among the most common conductors. Electrons have a negative charge and will normally sit around the positive protons of their own atom, but if a neighbouring atom has a positive charge (due to a missing electron) then the electron will ‘jump’ across. When the electron jumps it leaves behind a gap (a positive charge in that atom), this leaves room for another electron to fill that gap and so on. This gives the effect of the gaps (positive charge) moving in one direction, and the electrons moving in another.

Conventional current flows to the right, while the electrons flow to the left. This kind of current is called direct current or DC, which means the current only goes in one direction. There is another form of current called alternating current, AC, which has current alternating in direction it flows, this will be covered in a later tutorial.

S.I Units

Electronics has many different units, there are 4 primary units:

Q - Charge, measured in coulomb (C)

J - Joules, (1J is the amount of energy needed to raise one gram of water by 1 degree Celsius)

t - time, measured in seconds

From these we can derive the following units:

V - Voltage, also known as potential difference, measured in volts, which is equal to Jules per coulomb (J/c)

I - Current, measured in amps (A), which is equal to coulomb per second (C/s)

R - Resistance, measured in ohms ()

P - Power, measured in watts (W), which is equal to Jules per second (J/s)

In addition to these all the metric prefixes apply, for example 1kV = 1,000V and 1mA = 0.001A.

Formulas

There are 2 basic equations used in electronics

V=IR

P=IV

These 2 equations can be transformed to work out various things

that we need solutions to. For Example:

P=v^2/R

P=I^2R

R=P/I etc.

For resistors in parallel we have:

R­t = 1/R 1 + 1/R 2

where Rt is the total resistance and R1, R2 are the parallel resistors (this can also be extended to have R3, R4 etc.)

Circuits

There are many different electrical components, all with their own different functions. So that electronics can be recorded onto paper or computer we have to have symbols to represent all the components. Some basic components and their respective symbols are:

Resistor

-Dissipates electrical energy as heat.

Power supply

-Supplies a current and a volta ge to the circuit, for example a battery.

Light bulb

-Converts electrical energy to light and heat.

Ammeter

- An ammeter measures current in wire.

Voltmeter

- Measures the potential difference in voltage across two points.

All the components are connected via wires, which is just a line.

Here is an example of a basic circuit:

Rt = R1

The electrical resistance of this circuit is equal to the resistance of the

single component in it, in this case the resistor R.

If two resistors are placed in a circuit like so:

Rt = R1 + R2

Then the resistors are said to be in series, and the total resistance of the circuit is equal to the sum of the resistors.

There is another way to place the resistors in this circuit, like so:

Rt = 1/R1 + 1/R2

These resistors are said to be in parallel, now the total resistance of the circuit is summarised by the above equation.

Example 1

Find the total resistance of the following circuit:

First simplify the 10 and 20 resistors by considering them as one 30 resistor:

Now apply the equation: 1/Rt = 1/R1 + 1/R2

1/Rt = 1/20 + 1/30 :: substitute in values

1/Rt = 5/60 :: add the two fractions

Rt = 12 :: flip both sides of the equations over

This circuit has a total resistance of 12 Ohms.

Ammeters, Voltmeters & V=IR

Ammeters measure current, voltmeters measure volta ge. There is also such a device that can measure both (not simultaneously) called a multimeter. To work out the volta ge on a component the voltmeter must be parallel to the component, while if you want to measure the current through it you must place it in series.

The circuit shows a 100 Ohm resistor with 9 Volts potential across it. These nine volts are shown on the voltmeter. The ammeter shows that the resistor has 90mA running through it.

The relationship between these variable is summarised with the equation:

V=IR

Now if you are given any two of these quantities you can work out the third.

Example 2

Given the values for the current and voltage, what should the resistance of the resistor in this circuit be?

V=IR

9=0.036R Remember to convert mA to A

R=9/0.036 Divide both sides by .036

R=250 Ohms

Example 3

What does the Ammeter read?

First find the total resistance of the circuit (ammeter has negligible resistance)

Rt = 250 + 100 = 350 Ohms

V=IR

9=350I sub in values

I=9/350

I=0.0257A=25.7mA

Challenge Question

Find the current running through the ammeter and 250 Ohm resistor.

Answer: 36mA (highlight this line)

Power

Power is the amount of energy dissipated in a component per second it is measured in Watts (with units J/s). If a component uses 100J in 5 seconds it is running at 20W.

P=IV, P=V 2 /R also P=I 2 R

In a resistor this energy is dissipated as heat, if to much energy is put through a resistor then it will be fried, just like a light bulb burning out.

Example 4

A resistor with potential difference 2V across it and a current of 1 amp will have what power dissipated in it?

P=IV

P=1×2

P=2Watts

Example 5

What is the power dissipated in the 100 Ohm resistor?

Long Way

First find the total current flowing from the 12V power supply.

1/Rt = 1/100 + 1/200

1/Rt = 3/200

Rt = 66.7 Ohms is the total resistance of the circuit.

V = IR

12 = I*66.7

I = 12/66.7

I = 0.18 A ps

We now know the total current is 0.18 Amps, we also know that there is .06 Amps flowing through the ammeter and the 200 Ohm resistor, this leaves 0.12 Am ps to flow through the 100 Ohm resistor.

P = IV

P = 0.12*12

P = 1.44 Watts

Short Way

Using one of the above equations:

P = V 2 /R

P = 12 2 /100

P = 1.44 Watts

Now that one was much better wasn’t it! But there are some questions that will be designed to make this approach impossible, so it is useful to understand both methods.

That’s it for Basic Electronics! Thanks for reading.

admin Tutorials electronics, physics

Introduction

The Gauge principle is one of the most powerful principles in modern theoretical physics. It can be used to usher in the four fundamental forces of nature and potentially even unify them.

The term Gauge was originally used by Herman Weyl in an attempt to unify General Relativity (GR) with Electromagnetism.¹ Weyl noticed that GR was invariant under changes of ‘gauge’ or length scale. However while GR can be described geometrically, Electromagnetism cannot, so his attempt at unification failed. Weyl continued to use the word gauge to describe physical invariance even though it no longer had anything to do with length scales.¹

Gauge and Electromagnetism

A simple example of gauge invariance can be understood by considering an electrical circuit (figure 1)

Figure 1. Changing the potential globally does not alter the circuit behaviour.

By changing the potential in this circuit by 10 volts everywhere, the overall behaviour of the circuit has not changed. The circuit is said to be invariant with respect to global transformations to the electric potential.

Gauge invariance is thereby used to describe physical symmetries such as the universe’s invariance with respect to position in space time. This means if you perform a given experiment here today and in the vicinity of Betelgeuse a year from now, you will expect to get the same results. There are many examples of gauge invariance; here we will take a close look at Electromagnetism.

The theory of Electromagnetism describes two physical fields, the electric field E and the magnetic field B. From these two fields we can define associated potentials, the scalar potential and vector potential A, given by:

	(1)
	(2)

Since the E and B fields are obtained via derivatives of both and A, addition of constants to these potentials leaves the fields unchanged. This is called Gauge Freedom; there are an infinite amount of potentials that can represent the same field. This property can be exploited to make a choice of gauge (a particular and A) that simplifies calculations. Changing the potential by a constant is called a Gauge Transformation of the first kind (a global transformation). It is also possible to make Gauge Transformations of the form:

	(3)
	(4)

Where f = f(x,t) is a continuous function. This is called a Gauge Transformation of the second kind. Since f is a function of x (space) and t (time), this is now a local transformation that can vary over space and time. The potential can be transformed arbitrarily, as long as the new ‘gauge’ satisfies equations (1) and (2) of the original E and B fields. This may lead us to question the physicality of the potential; if the choice of representation is arbitrary then perhaps the potential is just a convenient mathematical artifice that describes the ‘physical’ field. However, this view is incorrect, as can be observed via the Aharonov-Bohm effect.

Figure 2. Setup of the Aharonov-Bohm effect.¹

The Aharonov-Bohm effect is the phenomenon of an electron diffraction pattern been shifted by what appears to be a non-local field. Starting with the normal double slit diffraction setup, a solenoid is placed between the two slits (see figure 2). With no current in the solenoid, the usual diffraction pattern is observed. When there is current flowing in the solenoid the electron diffraction pattern is shifted horizontally. Having a current in the solenoid induces a magnetic field on the interior of the coil, directed perpendicular to the plane of the page, and zero outside the coil. This appears to be action at a distance, as there is no local field to provide a force that shifts the electron diffraction pattern. In fact the vector potential outside the solenoid may be non-vanishing and merely curl-free, which will still give B = 0 via equation (1). Dirac showed that an electron travelling in a potential will pick up an extra phase factor, which changes the positions of the maxima and minima in the diffraction pattern. This effect experimentally verifies that it is incorrect to consider a potential being any less physical than the field it is derived from.

As introduced above, gauge transformations of the second kind are known as local transformations as they can vary in space and time. This type of transformation in intimately connected with the four fundamental forces of nature. Here we will consider in detail the electromagnetic force. Consider a particle with wavefunction ψ, which evolves in time according to the Schrödinger equation:

(5)

Where all symbols are as usually defined. While this describes the physical evolution of the wavefunction over time, what we actually observe is the probability density |² changing. It can be trivially shown that the observable probability of the wavefunction is invariant under transformations of the form:

(6)

Where θ is a constant and thus this is a global transformation. This changes the wavefunction’s phase by an amount θ. Under this transformation the wavefunction will still satisfy the Schrödinger equation. Yang and Mills questioned the validity of considering global transformations, since the speed of light prohibits instantaneous global transformations. This sent them down another path, what if instead θ = θ(x) (i.e. a local transformation is made)? The wavefunction would simply no longer satisfy the Schrödinger equation. This is because the Laplacian derivative will now include an extra term due to the transformation’s dependence on x. To maintain gauge invariance we must modify the momentum operator in the Schrödinger equation by introducing the covariant derivative²:

(7)

Where e is the charge on the particle and A_µ =_µθ. Using equation (7), we can re-write the Schrödinger equation with natural units as:

Where contains the coefficients of each of the 4-vector derivatives. Proceeding on by making the transformation:

Gives:

Making the substitution for A_µ and expanding:

Cancelling the terms leaves:

With e=1 (natural units) the Schrödinger equation is recovered:

As shown above, by using the covariant derivative in place of the momentum operator, will now solve the Schrödinger equation when under local phase transformations. This was at the cost of introducing a non-zero, which is the electromagnetic 4-vector potential: