Charge

a) Electric current as rate of flow of charge:

• The rate of flow of electric charge, and it’s typically denoted by the symbol I.
• – The formula you provided,
• [math] I = \frac{\Delta Q}{\Delta t} [/math]
• Represents the definition of electric current:
• – I is the electric current in amperes (A)
• – ∆Q is the amount of electric charge in coulombs (C) that flows through a given area
• – ∆t is the time in seconds (s) over which the charge flows
• 
• Figure 1 Electric current
• This formula states that electric current is equal to the amount of charge that flows through a given area per unit time. In other words, it measures how quickly electric charge is moving.
• Examples:
• (1)
• – If 2 coulombs of charge flow through a wire in 1 second, calculate the electric current.
• Given data:
• Charge flow in the wire = ∆Q = 2C
• Time = ∆t =1s
• Find data:
• Electric current =?
• Formula:
• [math] I = \frac{\Delta Q}{\Delta t} [/math]
  
• Solution:
• [math]\begin{gather}
  I &= \frac{\Delta Q}{\Delta t} \\
  I &= \frac{2}{1} \\
  I &= 2 \, \text{A}
  \end{gather} [/math]
  
• This means that the electric current is 2 amperes.
• It’s important to note that electric current can be either direct (DC) or alternating (AC), and it can flow through various materials, including wires, circuits, and even living organisms.
• (2)
• – A wire carries a charge of 20 Coulombs in 4 seconds. What is the electric current?
• Given data:
• Charge flow in the wire = ∆Q = 20C
• Time = ∆t =4s
• Find data:
• Electric current =?
• Formula:
• [math] I = \frac{\Delta Q}{\Delta t} [/math]
• Solution:
• [math]\begin{align*}
  I &= \frac{\Delta Q}{\Delta t} \\
  I &= \frac{20}{4} \\
  I &= 5 \, \text{A}
  \end{align*} [/math]
  
• This means that the electric current is 5 amperes.
• (3)
• – A battery delivers 500 Coulombs of charge in 10 minutes. What is the electric current?
• Given data:
• Charge flow in the wire = ∆Q = 500 C
• Time = ∆t =10 mint =10 × 60 = 600 s
• Find data:
• Electric current =?
• Formula:
• [math] I = \frac{\Delta Q}{\Delta t} [/math]
• Solution:
• [math] \begin{align*}
  I &= \frac{\Delta Q}{\Delta t} \\
  I &= \frac{500}{600} \\
  I &= 0.83 \, \text{A}
  \end{align*} [/math]
  
• This means that the electric current is 0.83 amperes.

b) The coulomb as the unit of charge:

• The coulomb (C) is the SI unit of electric charge. It is defined as the quantity of electricity transported in one second by a current of one ampere.
• One coulomb is equal to:
• – [math] 6.24 \times 10^{18} [/math]elementary charges (protons or electrons)
• – 1 ampere-second (A·s)
• – 1 volt-second (V·s) / ohm (Ω)
• The coulomb is used to measure:
• – Electric charge (Q)
• – Amount of electricity (quantity of charge)
• Examples:

(1)
A current of 2 A flows for 5 s. Find the charge.
Given data:
Current = I = 2 A
Time = ∆t =5 s
Find data:
Charge =?
Formula:

[math]∆Q = I∆t [/math]

Solution:

[math]∆Q = I∆t \\
∆Q = (2)(5) \\
∆Q = 10 C [/math]

• This means that the charge is 10 coulombs.

(2)
A device draws 0.5 A for 2 hours. Find the charge.
Given data:
Current = I = 0.5 A
Time = ∆t =2h =
Find data:
Charge =?
Formula:

[math]∆Q = I∆t [/math]

Solution:

[math]∆Q = I∆t \\
∆Q = (0.5)(7200) \\
∆Q = 3600 C [/math]

• This means that the charge is 3600 coulombs.
• The coulomb is a relatively large unit of charge, so smaller units like millicoulombs (mC), microcoulombs (μC), and nanocoulombs (nC) are often used in practice.

d) Net charge on a particle or an object is quantized and a multiple of e:

• The net charge on a particle or an object is indeed quantized and a multiple of the elementary charge (e).
• [math]\text{Elementary charge (e)} = 1.6 × 10^{-19} Coulombs (C)
• This means that the net charge on a particle or object can only be:
• 0, ±e, ±2e, ±3e, …
• In other words, the net charge is always an integer multiple of the elementary charge (e).
• Examples:
• – A proton has a charge of +e ([math]1.6 × 10^{-19} [/math]C)
• – An electron has a charge of -e ([math]-1.6 × 10^{-19} [/math]C )
• – A helium nucleus (alpha particle) has a charge of +2e ([math] 3.2 × 10^{-19}[/math] C)
• – A dust particle with 5 excess electrons has a charge of -5e ([math] 8 × 10^{-19}[/math] C)
• This quantization of charge is a fundamental aspect of physics and has important implications for our understanding of the behavior of particles and objects at the atomic and subatomic level.

e) Current as the movement of electrons in metals and movement of ions in electrolytes:

• Electric current can be understood as the movement of:
• Electrons in metals (electronic conduction):
• – Metals have a “sea” of free electrons that can move freely within the material.
• When a voltage is applied, these electrons flow through the metal, carrying electrical energy with them.
• Ions in electrolytes (ionic conduction):
• – Electrolytes are substances that dissolve in water and produce ions (charged particles).
• – When a voltage is applied, these ions move through the solution, carrying electrical energy with them.
• In both cases, the movement of charged particles (electrons or ions) constitutes an electric current.
• – In metals, the electrons flow from the negative terminal to the positive terminal (electron flow).
• – In electrolytes, the ions move in opposite directions: cations (positive ions) move towards the cathode (negative electrode), while anions (negative ions) move towards the anode (positive electrode).
• – The movement of charged particles is influenced by the electric field and the properties of the material or solution.

f) Conventional current and electron flow:

• Conventional Current:
• – Defined as the flow of positive charges (protons) from the positive terminal to the negative terminal
• – Also known as “Franklin’s convention” after Benjamin Franklin
• Electron Flow:
• – Defined as the flow of negative charges (electrons) from the negative terminal to the positive terminal
• – Opposite in direction to conventional current
• – Used in some physics and electronics contexts to emphasize the actual movement of electrons.
• 
• Figure 2 Conventional flow and electron flow
• Conventional current is a useful fiction, as it simplifies circuit analysis and is easier to work with mathematically
• Electron flow is the actual physical phenomenon, but it’s more cumbersome to work with mathematically
• The difference between conventional current and electron flow is important to understand in certain contexts, such as:
• Electronics: Understanding electron flow is crucial for designing and analyzing electronic circuits

g) Kirchhoff’s first law; conservation of charge:

• Kirchhoff’s First Law, also known as the Law of Conservation of Charge, states:
• “The total electric charge entering a node (or junction) is equal to the total electric charge leaving the node.”
• In other words, the law states that:
• [math]∑I_{in} = ∑I_{out}[/math]
• – Where is the current entering the node and  is the current leaving the node.
• 
• Figure 3 Kirchhoff’s first law
• This law is based on the principle of conservation of charge, which states that electric charge cannot be created or destroyed, only transferred from one place to another.
• Kirchhoff’s First Law has important implications for circuit analysis:
• – Charge is conserved: The law ensures that the total charge in a closed system remains constant.
• – Node analysis: The law allows us to analyze circuits by considering the currents entering and leaving each node.
• – Circuit simplification: The law enables us to simplify complex circuits by combining currents and reducing the number of nodes.

2. Mean drift velocity:

a) Mean drift velocity of charge carriers:

• The mean drift velocity of charge carriers is the average velocity of charged particles, such as electrons or holes, as they move through a conductor or semiconductor under the influence of an electric field.
• Drift velocity is defined as the net velocity of charge carriers gained due to the electric field, and it’s typically much smaller than the random thermal velocity of the carriers.
• The mean drift velocity is given by:
• [math]v_d = μE [/math]
• Where:
• [math]v_d[/math] = drift velocity (m/s)
• μ = mobility of the charge carriers (m²/Vs)
• E = electric field strength (V/m)
• 
• Figure 4 drift velocity of current
• The mean drift velocity is important in understanding:
• – Electric current: Drift velocity is directly proportional to current.
• – Conductivity: Mobility and drift velocity determine conductivity.
• – Semiconductor behavior: Drift velocity explains the behavior of charge carriers in semiconductors.
• – Device operation: Drift velocity is crucial in understanding device operation, such as in transistors and diodes.

b) I = Anev here n is the number density of charge carriers:

• I = Anev
• Where:
• – I is the current
• – A is the cross-sectional area
• – n is the number density of charge carriers (number of charge carriers per unit volume)
• – e is the elementary charge (charge of a single charge carrier)
• – v is the drift velocity (average velocity of charge carriers)
• This equation relates the current (I) to the number density of charge carriers (n), their charge (e), and their drift velocity (v).
• Number density (n) is a crucial parameter in calculating current, as it represents the number of charge carriers available to conduct electricity.
• Some examples:
• – Copper wire: [math] n ≈ 10^{29}[/math]electrons/m³
• – Silicon semiconductor: [math] n ≈ 10^{15} [/math]holes/m³ (in the valence band)
• By plugging in the values, you can calculate the current (I) using the equation
• [math]I = Anev [/math]
• This equation is fundamental in understanding electronic devices and circuits.

c) Distinction between conductors, semiconductors and insulators in terms of n:

• The distinction between conductors, semiconductors, and insulators can be understood in terms of the number density of charge carriers (n) as follows:
• Conductors:
• – High number density of charge carriers ([math]n ≈ 10^{28} –  10^{28} m^{-3} [/math])
• – Electrons are freely available for conduction
• – Examples: metals like copper, aluminum, gold
• Semiconductors:
• – Intermediate number density of charge carriers ([math]n ≈ 10^{15} –  10^{22} m^{-3} [/math])
• – Electrons require energy to become free for conduction
• – Examples: silicon, germanium, gallium arsenide
• Insulators:
• – Very low number density of charge carriers ([math]n ≈ 10^{0} –  10^{15} m^{-3} [/math] )
• – Electrons are tightly bound and require significant energy to become free for conduction.
• 
• Figure 5 Differences between insulators, semiconductors, and conductors
• – Examples: glass, ceramic, wood, plastic
• These are general guidelines, and the exact values of n can vary depending on the specific material and conditions.
• – Conductors have a high density of free electrons, making them suitable for conduction.
• – Semiconductors have a moderate density of free electrons, requiring energy to become conductive.
• – Insulators have a very low density of free electrons, making them poor conductors.
• This distinction is crucial in understanding the behavior of materials in various electronic and electrical applications.