Electric current
Electric current is the flow of charge carriers in a conductor.
When a potential difference is applied across a conductor, the free electrons in the conductor experience a force and start to move.
This movement of electrons constitutes an electric current.
The charge carried by the charge carriers is quantized, which means that the charge carried by each individual carrier can only take on certain discrete values. In most cases, the charge carriers in a conductor are electrons, which have a charge of
-1.6 x 10⁻¹⁹ C
The relationship between current, charge, and time is given by the equation.
Q=It
where Q is the charge, I is the current, and t is the time.
This equation tells us that the amount of charge that passes through a conductor is directly proportional to the current and the time for which the current flows.
For a current-carrying conductor, the current density is given by, I=Anvq Where: A is the cross-sectional area of the conductor, n is the number density of charge carriers, v is their drift velocity, and q is their charge.
This equation tells us that the current density depends on the cross-sectional area of the conductor, the number density of charge carriers, their drift velocity, and their charge.
The drift velocity is the average velocity at which the charge carriers move in response to an applied electric field, and it is typically much slower than the speed of light.
The number density of charge carriers is the number of charge carriers per unit volume of the conductor.
The charge q is the charge carried by each charge carrier, which is typically the charge of an electron in the case of metallic conductors.
Worked examples:
• A copper wire of length 2m and cross-sectional area 1mm² has a resistance of 0.1 Ω. What is the number density of conduction electrons in the wire? Assume that each copper atom contributes one free electron.
Solution
Resistance of the wire is given by R = ρL/A, where ρ is the resistivity of copper, L is the length of the wire, and A is the cross-sectional area.
So,
ρ = RA/L
= (0.1 Ω)(2 m)/ (1 mm²)
= 200 x 10⁻⁹ Ωm
The number density of conduction electrons, n, is given by,
n = N/V
where N is the total number of free electrons in the wire and V is the volume of the wire.
Since each copper atom contributes one free electron, the number of copper atoms per unit volume is equal to the atomic density of copper, which is 8.5 x 10²⁸ m⁻³.
The volume of the wire is given by.
V = AL
= (1 mm²)(2 m)
= 2 x 10⁻⁶ m³.
Therefore, the total number of free electrons in the wire is
N = (8.5 x 10²⁸ m⁻³) (2 x 10⁻⁶ m³)
= 17 x 10²².
So, the number density of conduction electrons in the wire is
n = N/V
= (17 x 10²²)/(2 x 10⁻⁶)
= 8.5 x 10²⁸ m⁻³.
Therefore, the number density of conduction electrons in the wire is 8.5 x 10²⁸ m⁻³.
Resistance and resistivity
Resistance:
Resistance is the measure of how much a material or device opposes the flow of electrical current through it.
It is caused by the collisions between free electrons and atoms in the material, which results in the conversion of electrical energy into heat energy.
Resistance is measured in ohms (Ω) and is represented by the symbol R.
Ohm's Law:
Ohm's law states that the voltage (V) across a conductor is equal to the current (I) flowing through it multiplied by its resistance (R).
Mathematically, this is expressed as V = IR
Examples:
A) If a circuit has a resistance of 5 Ω and a current of 2 A, what is the voltage across the circuit?
Solution
Using Ohm's law: V = IR = 2 A x 5 Ω = 10 V
Therefore, the voltage across the circuit is 10 volts.
B) If a circuit has a voltage of 12 V and a resistance of 3 Ω, what is the current flowing through the circuit?
Solution
Using Ohm's law: I = V/R = 12 V / 3 Ω = 4 A
Therefore, the current flowing through the circuit is 4 amperes.
C) A circuit consists of two identical resistors connected in parallel. The resistance of each resistor is R. If the total resistance of the circuit is 3 Ω, what is the value of R? (A) 1 Ω (B) 2 Ω (C) 3 Ω (D) 6 Ω
Solution
The circuit consists of two identical resistors connected in parallel. The resistance of each resistor is R. Let's call the total resistance of the circuit Rₜ.
When resistors are connected in parallel, the total resistance of the circuit is given by the formula:
1/Rₜ = 1/R₁ + 1/R₂ + 1/R₃ + ...
In this case, there are only two resistors, so the formula becomes:
1/Rₜ = 1/R + 1/R
Simplifying this expression, we get:
1/Rₜ = 2/R
To find the value of R, we can rearrange this expression as:
R = 2Rₜ
We also know that the total resistance of the circuit is 3 Ω. Substituting this into the equation above, we get:
R = 2 x 3 Ω = 6 Ω
Therefore, the value of R is 6 Ω.
The correct answer is (B).
Voltage/ current graph of a metallic conductor at constant temperature.
Voltage/ current graph of a semiconductor diode.
Voltage/ current graph of a lightbulb.
Filament lamp
The resistance of a filament lamp is directly proportional to its temperature.
As current flows through the filament of the lamp, the filament gets heated.
As the temperature of the filament increases, its resistance also increases.
This is due to the fact that the electrons in the filament gain energy as they collide with the atoms in the filament, making it harder for them to move through the material.
As a result, the current flowing through the lamp decreases, according to Ohm's law (V = IR).
Ohm's law
Ohm's law states that the current flowing through a conductor between two points is directly proportional to the voltage across the two points, provided that the temperature and other physical conditions remain constant.
Mathematically, Ohm's law can be expressed as:
I = V/R
where I is the current flowing through the conductor,
V is the voltage across the two points
R is the resistance of the conductor.
This equation implies that as the voltage across the conductor increases, the current through it also increases proportionally, as long as the resistance of the conductor remains constant.
Conversely, if the resistance of the conductor increases, the current flowing through it will decrease for a given voltage.
Examples:
A) A circuit has a current of 0.5A flowing through a resistor with a resistance of 100Ω. What is the voltage across the resistor?
Solution
V = IR
= 0.5A x 100Ω
= 50V
B) A circuit contains a resistor with a resistance of 12Ω and a battery with an emf of 9V and an internal resistance of 3Ω. Calculate the power delivered to the resistor by the battery.
Solution
To solve this problem, we first need to calculate the total resistance of the circuit. The total resistance is the sum of the resistance of the resistor and the internal resistance of the battery:
R_total = R_resistor + R_internal
R_total = 12Ω + 3Ω = 15Ω
Next, we can use Ohm's law to calculate the current in the circuit:
I = V / R_total
I = 9V / 15Ω
I = 0.6A
Finally, we can use the formula for electrical power to calculate the power delivered to the resistor by the battery:
P = I² x R_resistor
P = (0.6A) ² x 12Ω
P = 4.32W
Therefore, the power delivered to the resistor by the battery is 4.32 watts.
Resistivity
The formula
R = ρL/A
is used to calculate the resistance (R) of a conductor, where.
ρ is the resistivity of the material,
L is the length of the conductor
A is the cross-sectional area of the conductor.
Examples:
A) A copper wire has a length of 2 meters and a cross-sectional area of 1.5 mm². If the resistivity of copper is 1.68 x 10⁸Ωm, what is the resistance of the wire?
Solution
R = ρL/A
R = (1.68 x 10⁻⁸ Ωm) x (2 m) / (1.5 x 10⁻⁶ m²) R = 0.0224 Ω
B) An aluminum wire has a resistance of 5 Ω and a length of 4 meters. If the resistivity of aluminum is 2.65 x 10⁸ Ωm, what is the cross-sectional area of the wire?
Solution
R = ρL/A
A = ρL/R
A = (2.65 x 10⁻⁸ Ωm) x (4 m) / (5 Ω) A = 2.12 x 10⁻⁶ m²
C) A nichrome wire has a cross-sectional area of 0.2 mm² and a resistance of 8 Ω. If the resistivity of nichrome is 1.0 x 10^-6 Ωm, what is the length of the wire?
Solution
R = ρL/A
L = RA/ρ
L = (8 Ω) x (0.2 x 10⁻⁶ m²) / (1.0 x 10⁻⁶ Ωm) L = 1.6 m
D) A wire with a length of 1.5 m and a resistance of 0.05 Ω is made of a material with a resistivity of 2.5 x 10⁻⁸ Ωm. The wire is connected in series with a 15 V battery. What is the current in the wire?
Solution
Using Ohm's law, we know that:
V = IR
Rearranging this formula gives:
I = V / R
Substituting the given values:
I = 15 V / 0.05 Ω
I = 300 A
Therefore, the current in the wire is 300 A. E) A wire made of a certain material has a length of 2.0 m and a cross-sectional area of 0.5 mm². The resistivity of the material is 2.5 x 10⁻⁸ Ωm. The wire is connected to a battery with an electromotive force of 9.0 V. What is the current in the wire?
Solution
The formula for resistance of a wire is:
R = ρL/A
Substituting the given values:
R = (2.5 x 10⁻⁸ Ωm) x (2.0 m) / (0.5 mm² x 10⁻⁶ m²/mm²)
R = 0.2 Ω
Using Ohm's law, we know that:
V = IR
Rearranging this formula gives:
I = V / R
Substituting the given values:
I = 9.0 V / 0.2 Ω
I = 45 A
Therefore, the current in the wire is 45 A.
LDR
A light-dependent resistor (LDR) is a type of resistor whose resistance varies with the amount of light it is exposed to.
The resistance of an LDR is inversely proportional to the light intensity, meaning that as the light intensity increases, the resistance of the LDR decreases.
LDRs are used in a variety of applications to control the amount of current flowing through a circuit, allowing for automatic adjustment of the light intensity.
Thermistor
A thermistor is a type of resistor that has a temperature-dependent resistance.
The resistance of a thermistor decreases as its temperature increases.
This property is known as a negative temperature coefficient (NTC).
Thermistors are made of semiconducting materials, which means that their electrical properties change as their temperature changes.
At low temperatures, thermistors have a high resistance, while at high temperatures they have a low resistance.
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