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CURRENT ELECTRICITY
The branch of Physics that deals with the study of the motion of electric charges is known as current electricity.
Electric Current (I)
The rate of flow of electric charges through any cross-sectional area of a conductor is called electric current. If a net charge q passes through any cross section of a conductor in time t , then the current I is given by
Electric current 𝐼 =
𝑞 𝑡
where q is in coulomb and t is in second. Its SI unit is ampere (A).
1 𝑐𝑜𝑢𝑙𝑜𝑚𝑏 1 𝑠𝑒𝑐𝑜𝑛𝑑
1 ampere is defined as the current in a wire if one coulomb of charges flows through it in one second.
If the rate of flow of charge is not uniform, the current varies with time and the instantaneous value of current i is given by,
𝑑𝑞 𝑑𝑡
Current is a scalar quantity. The conventional direction of electric current is the direction of motion of positive charge or opposite to the direction of flow of electrons.
In a metallic conductor current flows due to motion of free electrons. In liquids like electrolytic compounds (HCl, HNO 3 , NaOH, H 2 SO 4 , CaCl 2 , KNO 3 etc.), positive and negative ions are the charge carriers. (Electrolytes are substances that dissociates into ions in solution and acquires the capacity to conduct electricity). In semiconductors, the charge carriers are electrons and holes. Gases are bad conductors. However, at very low pressures, if a high voltage is applied, then some particles become ionized resulting in negative and positive ions. Electrons and cations of ionized gases act as charge carriers.
Types of Electric Current
Electric current is of two types:
i. Direct Current (DC) - Its magnitude and direction do not change with time. A cell, battery or DC dynamo are the sources of direct current. ii. Alternating Current (AC) - An electric current whose magnitude changes continuously and changes its direction periodically is called alternating current. AC dynamo is source of alternating current.
Figure 1 (a) – DC current and voltage are constant in time, once the current is established.
Figure 1(b) – A graph of voltage and current versus time for 60-Hz AC power. The voltage and current are sinusoidal and are in phase for a simple resistance circuit.
Current Density
The electric current flowing per unit area of cross-section of a conductor is called current density.
Current density 𝐽 =
𝐼 𝐴
Its S1 unit is ampere metre-2^ and dimensional formula is [AL-2].
Current density is a vector quantity and its direction is in the direction of motion positive charge or in the direction of flow of current.
Under the influence of electric field, electrons move and subsequently collide with the ions. The time between two successive collisions is called Relaxation time given by 𝜏. The velocity of electron just before collision is given by
Where 𝑣 1 𝑎𝑛𝑑 𝑢⃗ 1 are the final and initial velocities and 𝜏 1 is the relaxation time of the electron.
The drift velocity is defined as the average velocity with which free electrons in a
conductor get drifted under the influence of an external field 𝐸⃗. It is represented by 𝑉⃗𝑑
Hence, 𝑉⃗𝑑 =
𝑣⃗ 1 +𝑣⃗ 2 +𝑣⃗ 3 +⋯+𝑣⃗𝑛 𝑁
𝑢⃗⃗ 1 +𝑢⃗⃗ 2 +𝑢⃗⃗ 3 +⋯+𝑢⃗⃗𝑛 𝑁
𝜏 1 +𝜏 2 +𝜏 3 +⋯+𝜏𝑛 𝑁
Where
𝜏 1 +𝜏 2 +𝜏 3 +⋯+𝜏𝑛 𝑁
Drift velocity 𝑉⃗𝑑 =
−𝑒𝐸⃗ 𝜏 𝑚
−𝑒𝑉𝜏 𝑚𝑙
𝑒 𝐸⃗⃗⃗ 𝜏 𝑚
𝑒 𝑉 𝜏 𝑚 𝑙
where, τ = relaxation time, e is charge on electron, 𝐸⃗ is electric field intensity, l is the length of the conductor, V is potential difference across the ends of the conductor.
Relation between electric current and drift velocity
Consider a conductor of cross section area A and length l. The volume of the conductor is given by
V = A l
If n be the number of free electrons per unit volume, then the total charge on all the free electrons is given by,
q = n A l e
under the influence of electric field, the electrons will travel the distance l with drift velocity V d in time t given by
𝑙 𝑉𝑑
From 𝐼 =
𝑞 𝑡
⟹ 𝐼 = 𝑛 𝐴 𝑒 𝑉𝑑 or 𝑉𝑑 =
𝐼 𝑛 𝐴 𝑒
Mobility
The drift velocity of electron acquired per unit electric field applied is known as the mobility of electron.
Mobility of electron (μ) is given by,
𝑉𝑑 𝐸⃗ Its SI unit is m^2 s-1V-1^ and its dimensional formula is [M-1T^2 A].
Ohm’s Law
If physical conditions of a conductor such as temperature remains unchanged, then the electric current (I) flowing through the conductor is directly proportional to the potential difference (V) applied across its ends.
I ∝ V or V = IR
where R is the electrical resistance of the conductor
Electrical Resistance
The obstruction offered by any conductor in the path of flow of current is called its electrical resistance.
Electrical resistance, 𝑅 =
𝑉 𝐼
Its SI unit is ohm (Ω) and its dimensional formula is [ML^2 T-3A-2].
Also, electrical resistance of a conductor 𝑅 =
𝜌𝑙 𝐴
where, l is the length of the conductor, A is the cross-section area and ρ is the resistivity of the material of the conductor.
Electrical resistance can also be expressed as,
Resistivity of metals increases with increase in temperature as
where 𝝆𝒐 and 𝝆𝒕 are resistivity of metals at 0° C and t° C. α is the temperature coefficient of resistivity of the material.
Metals have positive temperature coefficient of resistance, i.e., their resistance increases with increase in temperature.
For insulators, semiconductors and alloys:
Insulators and semiconductors have negative temperature coefficient of resistance, i.e., their resistance decreases with increase in temperature. The temperature coefficient is low for alloys. For some alloys like nichrome, manganin and constantan, α is positive but very low.
(In magnetic field the resistivity of metals increases. But resistivity of ferromagnetic materials such as iron, nickel, cobalt etc. decreases in magnetic field.)
Electrical Conductivity
The reciprocal of resistivity is called electrical conductivity.
Electrical conductivity (σ) is given by
1 𝜌
𝑙 𝑅𝐴
n 𝑒^2 𝜏 𝑚
Its SI units is ohm-1^ m-1^ or Ω-1^ m-1^ or Siemens m-1^ (S/m).
Relation between current density (J) and electrical conductivity (σ) is given by
n 𝑒^2 𝜏 𝑚
where, E = electric field intensity.
Ohmic Conductors and Non-ohmic Conductors
Those conductors which obey Ohm’s law, are called ohmic conductors. In other words there is a linear relationship between voltage and current for all values e.g. metals.
Those conductors which do not obey Ohm’s law, are called non-ohmic conductors. In other words the relationship between voltage and current is not linear for all values. e.g., diode valve, triode valve, transistor, vacuum tubes etc.
For ohmic conductors V – I graph is a straight line. For non-ohmic conductors V – I graph is not a straight line.
Superconductors
When few metals are cooled, then below a certain critical temperature their electrical resistance suddenly becomes zero. The ability of certain metals, their compounds and alloys to conduct electricity with zero resistance at very low temperatures is called superconductivity. The materials which exhibit this property are called superconductors.
The phenomenon of superconductivity was first observed by Kammerlingh Onnes in
- He found that mercury suddenly showed zero resistance at 4.2 K. The first theoretical explanation of superconductivity was given by Bardeen, Cooper and Schrieffer in 1957 and it is called the BCS theory.
The temperature at which electrical resistivity of the material suddenly drops to zero and the material changes from normal conductor to a superconductor is called the transition temperature or critical temperature TC. At
Mercury become superconducting at 4.2 K, lead at 7.19 K and niobium at 9.26 K
Applications of superconductors i. Superconductors form the basis of energy saving power systems, namely the superconducting generators, which are smaller in size and weight, in comparison with conventional generators.
2. In Parallel
Equivalent resistance
1 𝑅𝐸
1 𝑅 1
1 𝑅 2
1 𝑅 3
𝑉 𝑅 1
𝑉 𝑅 2
𝑉 𝑅 3
Potential difference across each resistor is same.
Sum of electric currents flowing through individual resistors is equal to the electric current drawn from the source.
Electric Cell
An electric cell is a device which converts chemical energy into electrical energy.
Electric cells are of two types
i. Primary Cells Primary cells cannot be charged again. Voltaic, Daniel and Leclanche cells are primary cells. ii. Secondary Cells Secondary cells can be charged again and again. Acid and alkali accumulators are secondary cells.
Recall that, just as water requires a pressure difference to start flowing, electrons require an electric potential difference to make them move. The potential difference provides the energy to create movement. Electric potential difference is also called voltage and it is measured in volts (abbreviated V ). In the case of water, pressure can be created by a water pump or difference in height, like a water tower. In electronics, batteries and electric generators are the common sources of voltage. The presence of two different charges also creates a voltage ; it gives the electric charges the energy to flow. A battery is a device consisting of one or more electrochemical cells with external connections for powering electrical devices.
Electro – motive – Force ( emf ) of a Cell
In order to maintain a potential difference between two points in the presence of a current, there must be a non-electrical source of energy replenishing the energy lost by the charges moving through that potential difference. The energy supplied/unit charge by this source is called the emf , whether the means for providing the emf is
chemical, magnetic, mechanical or any other process. The symbol used for an emf is 𝜀
The energy given by a cell in flowing unit positive charge throughout the circuit completely one time, is equal to the emf of a cell.
emf of a cell (𝜀) = W / q.
Its SI unit is volt.
Comparison of emf and potential difference
- The difference of potentials between the two terminals of a cell in an open circuit is called the electromotive force ( emf ) of a cell. The difference in potentials between any two points in a closed circuit is called potential difference.
- The emf is independent of external resistance of the circuit, whereas potential difference is proportional to the resistance between any two points.
Electric energy and electric power.
If I is the current flowing through a conductor of resistance R in time t , then the
quantity of charge flowing is, q = It. If the charge q , flows between two points having a potential difference V , then the workdone in moving the charge is
𝑊𝑜𝑟𝑘𝑑𝑜𝑛𝑒 = 𝑉𝑞 = 𝑉𝐼 𝑡
Then, electric power is defined as the rate of doing electric work.
Fig – Conduction in liquid
Faraday’s laws of electrolysis
The factors affecting the quantities of matter liberated during the process of electrolysis were investigated by Faraday.
First Law:
The mass of a substance liberated at an electrode is directly proportional to the charge passing through the electrolyte.
If an electric current I is passed through an electrolyte for a time t , the amount of
charge (q) passed is I t. According to the law, mass of substance liberated (m) is
𝑚 𝛼 𝑞 or 𝑚 = 𝑧𝐼𝑡
where Z is a constant for the substance being liberated called electrochemical equivalent. Its unit is kg C–^1.
The electrochemical equivalent of a substance is defined as the mass of substance liberated in electrolysis when one coulomb charge is passed through the electrolyte.
Second Law:
The mass of a substance liberated at an electrode by a given amount of charge is proportional to the *chemical equivalent of the substance.
If E is the chemical equivalent of a substance, from the second law
𝑚 𝛼 𝐸
Chemical equivalent =
𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑎𝑡𝑜𝑚𝑖𝑐 𝑚𝑎𝑠𝑠 𝑣𝑎𝑙𝑒𝑛𝑐𝑦 =
𝑚𝑎𝑠𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑡𝑜𝑚 1 2 𝑜𝑓 𝑡ℎ𝑒 𝑚𝑎𝑠𝑠 𝑜𝑓 𝐶
(^12) 𝑎𝑡𝑜𝑚 𝑥 𝑣𝑎𝑙𝑒𝑛𝑐𝑦
