Electric Currents and Basic Circuits

Question

Electric current (I):

Electric current is the flow of electric charge through a conductor over a period of time.

It is defined as the rate at which charge flows past a given point in a circuit.

Mathematically, it is expressed as:

i&#160;=&#160;qt&#160;(C/s)

Where;

q = charge (C)
t = time (s)

The unit of electric current is Coulomb per second or Amperes.

Potential difference or Voltage (V)

The potential difference between two points is the work done in moving a unit positive charge from one point to another in an electric circuit.

It is also called voltage, named after Alessandro Volta, the Italian physicist who invented the electric battery.

Mathematically,

V&#160;=&#160;WQ&#160;(joules/coulomb)

Where:

V = potential difference (volts)
W = work done (joules)
Q = charge (coulombs)

Its unit is expressed as volts or joules/coulomb.

Resistance

Resistance is a property of a material or component that describes its opposition to the flow of electric current in a circuit.

To understand resistance, imagine a tap controlling the flow of water from a tank. The more you close the tap, the less water flows out. If you fully close it, water stops flowing. This is similar to electrical resistance, the higher the resistance in a circuit, the less current flows. If the resistance becomes extremely high, current flow can become negligible or stop entirely.

Resistance in Conductors and Insulators

Conductors generally have low resistance, allowing electric current to flow easily. As discussed in our previous lecture, examples include metals, acid solutions, and salt solutions. Among metals, silver, copper, and aluminum are the best conductors of electricity, in that order.

On the other hand, insulators have high resistance and do not conduct electricity under normal conditions. They may only allow current to pass when exposed to very high temperatures.

Interestingly, the effect of temperature on resistance differs between conductors and insulators:

In conductors, increasing the temperature increases resistance.

In insulators, increasing the temperature decreases resistance.

Summary:

Conductors &rarr; High temperature &rarr; High resistance

Insulators &rarr; High temperature &rarr; Low resistance

Unit of resistance

The unit of resistance is ohms (&Omega;), named after George Simon Ohm.

For insulators whose resistances are very high, a much bigger unit is used i.e., mega-ohm = 106 ohm (the prefix &lsquo;mega&rsquo; or mego meaning a million) or kilo-ohm = 103 ohm (kilo means thousand).

In the case of very small resistances, smaller units like milli-ohm = 10&minus;3 ohm or micro-ohm = 10&minus;6 ohm are used.

Prefix

Its Meaning

Abbreviation

Equal To

Mega-

One million

M&Omega;

10⁶ &Omega;

Kilo-

One thousand

k&Omega;

10&sup3; &Omega;

Centi-

One hundredth

c&Omega; (rarely used)

10⁻&sup2; &Omega;

Milli-

One thousandth

m&Omega;

10⁻&sup3; &Omega;

Micro-

One millionth

&mu;&Omega;

10⁻⁶ &Omega;

Law of Resistance

Consider a conductor of length L, having a cross-sectional area A as shown below.

The resistance R of the conductor is:

Directly proportional to its length L

Inversely proportional to its cross-sectional area A

This relationship is expressed mathematically as:

R&#8733;LAR=&#961;LA

Where,

R = resistance (in ohms, &Omega;)
𝜌 = resistivity of the material (in &Omega;-m)
𝐿 = length of the conductor (in meters, m)
𝐴 = cross-sectional area (in square meters)

Example

Calculate the resistance of 100 m length of a wire having a uniform cross-sectional area of 0.1 mm2 if the wire is made of manganin having a resistivity of 50 &times; 10&minus;8 &Omega;-m.
	
	L = 100m, A = 0.1 mm2 = 0.1 * 10 -6, 𝜌
	R=&#961;LA&#160;=&#160;50&#215;10-8&#215;1000.1&#215;10-6=500&#937;
	 
	The resistance of a conductor 1 mm2 in cross-section and 20 m long is 0.346 &Omega;. Determine the specific resistance of the conducting material 
	
	L = 20m, R = 0.346 &Omega;, A = 1mm2 = 1 x 10-6m2
	&#961;=RAL&#160;=&#160;0.346&#160;*&#160;1&#160;*&#160;10-620=1.73&#160;*&#160;10-8&#937;-m

Ohms Law

Ohm&rsquo;s law states that the current flowing through a conductor is directly proportional to the voltage across it, provided the temperature and physical conditions (such as temperature), remain constant.

Mathematically,

V = IR

Where;
V = Voltage
I = Current.

Example

Calculate amount of current that flows through a resistor of resistance 40 ohms if the voltage across the resistor is 5 volts.

V = 5v, R = 40 &Omega;;

I&#160;=&#160;VR&#160;=540&#160;=&#160;0.125&#160;A

Resistors in Circuits

A circuit containing more than one resistor can be connected in three main ways:

Series connection
	Parallel connection
	Combination (Series-Parallel or Parallel-Series)

Resistors in Series

In a series connection, resistors are connected end-to-end in such a way that each terminal of a resistor is directly connected to the terminal of the next resistor, without any branching or node between them.

The same current flows through all resistors.
	The total resistance is the sum of individual resistances: R1

Rtotal= R1+R2+R3&hellip; Rn

Where R is the effective resistance of the series connected resistors.

Example

Calculate the effective resistance of the following resistors connected in series, 5 ohms, 9 ohms, and 10 ohms.
R = R1 + R2 + R3 = 5 + 9 + 10 = 24 ohms.

Resistors in Parallel

When resistors are connected in parallel, they share a common node. In a parallel connection, resistors are connected such that both terminals of each resistor are connected to the same two points&mdash;forming branches. Each resistor provides an independent path for current to flow.

&middot;       The voltage across all resistors is the same.

&middot;       The total current is the sum of the currents through each resistor.

V = V1 = V2 = V3
I = I1 + I2 + I3
But V = IR, I = V/R
Thus,
I1 = V1 / R1 , I2 = V2 / R2 , I3 = V3 / R3
Substituting, we have that
   V R  =   V 1     R 1    +   V 2     R 2    +   V 3     R 3&#8203;&#8203;&#8203;&#8203;

V R  =V   1   R 1    + 1   R 2    + 1   R 3

1 R  = 1   R 1    + 1   R 2    + 1   R 3