Current Electricity is one of the most important chapters in JEE Mains Physics. Every year, around 3 questions which bear a total of 12 marks are asked from this chapter. Therefore, Current Electricity has around 10% weightage in JEE Mains paper. Some of the topics tested under this section include electric current, drift velocity, electrical resistance, and Ohm’s law.  Check JEE Main Physics Syllabus

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JEE Main 2020 January session posed two questions on Current Electricity in each shift. Before taking a look at the study notes for Current Electricity, here is a question from the previous year’s question paper:

Suppose a wire is stretched to make it 0.1% lengthier, its resistance will - 

a. Decrease by 0.2%

b. Decrease by 0.05%

c. Increase by 0.05%

d. Increase by 0.2%

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If you are facing any difficulty in attempting questions from Current Electricity, refer to the study notes given below. 

Must Read:

• JEE Main Mathematics Preparation Tips
• Maximize Your JEE Main 2020 Score by Attempting Questions the Right Way

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What is Current Electricity? 

• Current Electricity is the rate of charge flow through any cross-section, i.e. I = dq/dt.
• The direction of the movement of positive charge is conventionally considered as the direction of electric current. It is a scalar quantity and the S.I. Unit is Ampere (A). 
• Current conductor carriers are electrons, (valence e – s) electrolyte ions, semiconductor electrons and holes, and positive ions or electrons in gases. 
• Charge of electron = 1.6 × 10–19c
• 1 ampere = 6.25 × 1018 electrons/sec
• While the direction is consistent with current (contradictory to electrons movement), it is not a vector quantity because it does not obey vector addition laws. 
• For a current to flow through a cross-section, a net charge flow across that cross-section must occur. There are about 1028 free electrons per m3 in a metal like copper, moving randomly in all directions with speeds of the order of 106 m/s even with the lack of electric field. But, because the number of electrons passing through a cross-section from left to right is equal to the number of electrons traveling in a given time from right to left, the total charge flow is zero and thus, the electrical current is zero. 
• As current flows through it, a conductor stays uncharged. The net charge in a conductor carrying current is zero.

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Ohm’s Law and Electrical Resistance 

• When a potential difference is seen across the extremities of a conductor, a current I gets formed in the conductor.
• According to Ohm’s law, “After keeping the given physical conditions like temperature, mechanical strain, etc. constant, the current (I) formed in the conductor is directly proportional to the potential difference (V) seen across the conductor”, i.e., or I=KV... (1)

where K is a constant of proportionality known as the conductance of the specific conductor.

• Alternatively, V = RI ... (2)

where the constant R is known as the electrical resistance or simply resistance of the specific conductor.

• From the given two equations. it is clear that R = 1/K.
• If a substance adheres to Ohm’s law, then there is an existence of a linear relationship between V and I as shown in figure 1. These substances are known as Ohmic substances. Some substances do not adhere to Ohm’s law. These are known as non-Ohmic substances as shown in figure 2.
• A few examples of non-ohmic conductors are diode valve, triode valve and electrolytes, and thermistors.

• The slope of V-I Curve of a conductor gives the resistance of the conductor, i.e. Slope = tan θ = V/I
• The SI unit of resistance R is volt/ampere = ohm (Ω)

Electrical Resistance 

Upon introduction of potential differential over a conductor's ends, the conductor's free e–s begin to migrate towards the conductor's positive end. They create collisions with the conductor's ions or atoms when drifting and thus, their mobility is obstructed. The net impediment to the movement of free e–s or simply current provided by a conductor is called electrical resistance. This depends on the conductor's size, geometry, temperature, and nature.

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Joule’s Law of Heating

Joule’s Law of Heating states that the quantity of heat produced in a conductor is directly proportional to the

• square of the current flowing through the conductor, (q, T – constt) i.e. H ∝ i2
• resistance of the conductor (i, T – constt.)

i.e. H ∝ R

• time for which the current is passed (i, R, – constt)

i.e., H ∝ t

Thus, H = i2 RT joule = i2 RT/4.2 cal.

Example: How much heat is produced by a 2 kW electric heater when it is operated for 30 minutes? 

Answer: Heat Produced = Rating in kW x (times of operation)

= 2 kW0.5 hour = 1 kWh

= 36×105joules or 860 kcal

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Kirchoff’s Law and Electrical Circuits 

Several practical combinations of resistors cannot be reduced to basic series and parallel combinations. For example, the resistors in the figure given below are neither in series nor in parallel combinations.

The usage of Ohm’s law is not enough to solve these problems. Kirchoff’s laws are utilized in such cases.

• In a circuit, a junction is a point where three or more conductors join.
• A loop is a closed conducting path.
• In the figure given above, e, f, d, c are junctions. a, b, are not junctions.
• The various loops are efde, cdfc, eabcf and eabcde.

Kirchoff’s Junction Law

This law is based on the conservation of charge. At any junction, the total sum of currents entering the junction should be equal to the total sum of currents leaving from it. If this is not the case, charges will pile up at the junction. This cannot occur because this would mean high or low potential is maintained at a point in a wire lacking any external influence.

When we use this rule at junction c, we will get I = I1 + I2

Kirchoff’s Loop Law

This law is based on energy conservation. The algebraic sum of changes in the potential around any closed loop of the circuit should be zero.

Sign convention for using loop law: Suppose we move a loop element (emf device, resistor, capacitor, or inductor) in the direction of rising potential, the potential difference will be taken as positive and vice-versa.

Problem-solving tactics for the usage of Kirchoff’s Law

• Make a circuit diagram that is large enough to depict all resistors, emf devices, capacitors, and currents clearly.
• Take into consideration the resistance of voltmeter, ammeter, or internal resistance of a cell.
• Assume the direction of current in all branches. It could be noted that one branch has just one direction of current. Use junction law simultaneously while creating currents. This will help in reducing the number of unknown quantities.

In the circuits given above, it is assumed that the direction of current I1 in branch abcd is anti-clockwise and the direction of current I2 in-branch afed is clockwise.

In figure 1, there are two unknown currents (I1, I2) whereas in figure 2, there are three unknown currents (I1, I2 and I3). Figure 1 is a better choice for solving problems. The junction rule can be used at d simultaneously when labelling currents.

• In a branch holding a capacitor, the current is zero when d.c is given and stable state conditions are received.
• There is now a need for as many independent equations as there are unknown conditions. If we have to discover a specific unknown, we must ensure that the unknown shows in one of the equations created.
• For creating equations, select the loop and fully traverse the loop. It could be travelled in clockwise or anticlockwise directions. When using the second law, utilize sign conventions appropriately.
• Solve the equations created to find the unknown conditions. Suppose any value of the current comes negative, then that particular current is in the opposite direction to the one assumed.

Applications

Use the second law in the loop abcda of figure 1 by taking the loop in an anti-clockwise direction beginning from a.

+ E2 – I1R4 – (I1 + I2) R3 = 0

For loop afeda, when moved in the clockwise direction, we get –

E1 – I2 R1 – I2R2 – (I1 + I2)R3 = 0

Node Method to Apply Kirchoff’s Law (Open Loop Method)

Step 1 - Choose a reference node and assume that its potential is (zero/x)V.

Step 2 - Calculate the voltage of the other selected points with reference to the reference node.

Step 3 - Find an independent node whose voltage is unknown. Apply Kirchoff’s law to obtain the relevant values.

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Current Density and Electric Power ​

Current Density

Current density at a point within a conductor is described as the quantity of current flowing around the point of the conductor per unit cross-sectional area, given the field is kept in a normal direction to the current direction, i.e - Current Density = I/A.

If the field is not normal to current, the field normal to current is A' = A cos θ (see figure). 

J = I/A cos θ or I = J A cos θ 

or

The SI unit is Am–2. The Current Density can also be associated with the electric field as -

where σ is the substance’s conductivity and ρ is the substance’s specific resistance.

J is a vector quantity and its direction is similar to that of .

The dimensions of J are [M°L–2T°A]

Electric Power

• Electric current is a physical macroscopic quantity while a microscopic physical quantity is the current density. 
• For a specified conductor, current with cross-sectional area change does not change.
• Electric Power is defined as the rate at which work is done to maintain the current in an electric circuit, i.e. Electric power, P = VI = I2R = V2/R watt or joule/second.
• The Electric Energy used up in a circuit is defined as the total work done to maintain the current in an electric circuit for a specified time period, i.e. Electric energy = VIt = Pt = I2 Rt = V2 t / R.
• The SI unit of electric energy is joule (denoted by J), where 1 joule = 1 watt × 1 second = 1 volt × 1 ampere × 1 sec.
• In household circuits, the electrical appliances are joined in parallel and the electrical energy used up is measured in kWh (kilowatt hour).
• 1 kWh (1 B.O.T. unit) = 1000 Wh = 3.6 × 106 J

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Drift Velocity 

When a conductor's ends are attached to the two terminals of a battery, an electric field from the positive terminal to the negative terminal is built up in the conductor. The free electrons in the conductor feel a force perpendicular to the course of the electric field and are accelerated accordingly. But, this acceleration process is quickly disrupted by collision with ions of solid. The average time each electron is accelerated prior to a collision is called the mean free time, or the mean relaxation time. 

Therefore, in addition to their unpredictable motion, the free electrons inside the metal gain a small velocity at the positive end of the conductor. This velocity is called Drift Velocity, i.e - , where e is the charge and m is the mass of the electron. 

 is the electric field set in the conductor and the average relaxation time.

The negative sign exists because the directions of and for the electron are opposite in nature. 

E = V/l

Here, V is the potential difference at ends of the conductor with the length l. The uniform current I, which flows through the conductor is given by -

I = n e A vd, where n = the number of free electrons per unit volume, A = area of cross-section, and vd = drift velocity. 

In vector form,

The negative sign exists because the direction of drift velocity of electron opposes .

Mobility - The Drift Velocity per unit electric field is known as mobility. It is denoted by µ.

The S.I. unit is m2/volt-sec.

• Drift velocity is very less. It has an order of 10–4m/s, which is negligible in comparison to the thermal speed of e–s at room temperature (105 m/s).
• The drift velocity is given by

where, J = current density , e = electronic charge = 1.6 × 10–19 C, and n = the number of free electrons per unit volume. 

• The number of free electrons per unit volume (n) can be known through the following relation:

and

where N0 = Avogadro number, d = density of the metal, M = molecular weight, and x = number of free electrons per atom.

• For stable current:

; ;

This shows that for a specified material and steady current in the situation of a non-uniform cross-section of material,

; ;

• Variation of drift velocity:

; Vd ∝ E

when the length is doubled, vd reduces to half, and when V is doubled, vd increases by two times. 

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Resistivity, Conductivity, and Material & their Resistivity 

Resistivity

For a specific conductor of uniform cross-section A and length l, the electrical resistance R is directly proportional to length l and inversely proportional to cross-sectional area A.

i.e., or or

ρ is known as the specific resistance or electrical resistivity.

Also,

The SI unit of resistivity is ohm - m.

Conductivity

Conductivity is described as the reciprocal of resistivity i.e. .

• The SI unit of conductivity is Ohm–1m–1 or mho/m.
• Ohm’s law could even be expressed as, J = σE

where J = current density and E = electric field strength.

• Conductivity, where n is the free electron density, τ is relaxation time, and m is mass of the electron.
• The value of ρ is very less for a conductor, very high for insulators and alloys, and in between that of conductors and insulators lies the value for semiconductors.
• Resistance is the feature of an object while resistivity is the feature of the material.

Materials and their Resistivity 

Effect of Temperature on Resistance and Resistivity: The resistance of a conductor is given as Rt = R0 (1 + αΔt), where α = temperature coefficient of resistance and Δt = change in temperature. 

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Metallic Conductors, Alloys, and Semiconductors

Metallic Conductors

Suppose ρ1 and ρ2 are resistivities of a conductor at temperatures t1 and t2, then ρ2 = ρ1 (1 + α Δ T)

where α = temperature coefficient of resistivity and

where ΔT = t2 – t1 = change in temperature

The value of α is positive for all metallic conductors. ∴ ρ2 > ρ1

In simpler words, with a rise in temperature, the positive ions of the metal vibrate with higher amplitude and they block the path of electrons more often. Owing to this, the mean path reduces and the relaxation time also lessens. This leads to a rise in resistivity.

Note that the value of α for most of the metals is

Alloys

The rate at which the resistance changes with temperature for alloys is less when compared to pure metals.

For example, an alloy manganin has a resistance 30-40 times that of copper for similar dimensions. The value of α for manganin is also very small ≈ 0.00001°C–1. Because of the above properties, manganin is useful in making wires for standard resistance in heaters, resistance boxes, and the like.

Note that eureka and constantan are the other alloys for which ρ is quite high. They are used to identify small temperatures and safeguard picture tubes or windings of generators and transformers.

Semiconductors

The resistivity of semiconductors drops with a rise in temperature. For semiconductors, the value of α is negative.

With a rise in temperature, the value of n also rises. Note that ρ decreases with a rise in temperature. However, the value of the increase in n dominates the value of ρ in this scenario.

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Electrolytes, Insulators, and Superconductors

Electrolytes

The resistivity of electrolytes falls with a rise in temperature. This is due to the fact that the viscosity of electrolytes drops with an increase in temperature for ions to get more freedom to move.

Insulators

The resistivity of insulators rises almost exponentially with a fall in temperature. The conductivity of insulators is close to zero at 0 K.

Superconductors

There are a few specific materials for which the resistance becomes zero below a particular temperature. This temperature is known as the critical temperature. Below the critical temperature, the material provides no resistance to the flow of e–s. The material here is known as a superconductor. The reason for superconductivity is that the electrons in these superconductors are mutually coherent and not mutually independent. This coherent cloud of e–s creates no collision with the ions of superconductors and hence, no resistance is given to the flow of e–s

For example, R = 0 for Hg at 4.2 K and R = 0 for Pb at 7.2 K. These substances are known to be superconductors at that critical temperature.

Superconductors are used

• to make very strong electromagnets
• to produce extremely high-speed computers
• to transmit electric power
• to study high energy particle physics and material science

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Series and Parallel Combination of Resistors

Resistance in Series

When a number of resistances are connected end to end such that current flows through each of the resistors upon some potential difference being given across the combination, the conductors are described to be connected in series.

The equivalent resistance in the series is denoted by

(Req)s = R1 + R2 + ...+ Rn

The equivalent resistance of the same resistances connected in series is always higher than the highest of individual resistances.

Potential division rule in series combination

Resistance in Parallel

The number of resistors is described to be connected in parallel when the same potential difference is seen across all resistors.

The equivalent resistance is denoted by

The equivalent resistance in a parallel combination is always lower than the value of the lowest individual resistance in the circuits.

Current division rule in parallel combination

;

In a specified combination of resistors, if you aim to detect whether the resistances are in series or in parallel, check if the same current flows through two resistors. If yes, then these are in series, and if the same potential difference is seen across two resistors, then these are in parallel. 

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Series and Parallel Combination of Cells 

Series Combination of Cells 

Equivalent Emf

EAB = E1 + E2 + ... + En

Equivalent internal resistance,

RAB = r1 + r2 + ....... + rn

Parallel Combination of Cells

Equivalent emf

Equivalent internal resistance

Mixed Grouping of Cells

Equivalent emf EAB = nE

Equivalent resistance =

Where n = no. of cells in a row, and M = no. of rows.

If this equivalent cell is connected to an external resistance R then,

• The precondition for maximum current to pass through external resistance R

⇒ R = nr/m

In simpler words, when external resistance is equal to the total internal resistance of all the cells,

the maximum current

• The maximum power dissipation for the circuit shown below.

Power

For maximum power across the resistor dP/dR=0,

On solving, the result is - R = r

This is the necessity for maximum power dissipation.

• Suppose identical cells are joined in a loop in order, the emf between any two points in the loop would be zero.
• Suppose n identical cells are joined in series and m are wrongly connected then,
• Enet = nE – 2mE

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Electromotive Force and Internal Resistance of a Cell

Electromotive Force

An electromotive force (emf) device consists of a positive terminal at a high potential and a negative terminal at a low potential. This device is responsible for carrying a positive charge within itself from the negative terminal to the positive terminal.

In order to make this happen, the work is done by an agency in the emf device. The energies needed to do this work are - chemical energy (like in a battery), mechanical energy (like that in an electric generator), and temperature difference (like in a thermopile).

The emf is hence given by the formula, E=dW/dq

• The S.I unit of emf is (V) J/C=vdt
• Electromotive force is a potential difference and not a force.
• It can be defined as the work completed in moving a charge around a closed circuit one time.

Internal Resistance

The potential difference over a true source of emf is not equal to its emf. This is because the charge which is passing inside the emf device also suffers resistance. This resistance is known as the internal resistance of the emf device.

E = IR + Ir = V + Ir

⇒ V = E – Ir

• For a cell

• Emf is the feature of a cell but terminal potential difference is dependent on the current pulled from the cell.

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Wheatstone Bridge and Meter Bridge or Slide Wire Bridge 

Wheatstone Bridge

The requirement for a balanced Wheatstone bridge is given below.

also

When the battery and galvanometer of a Wheatstone bridge is interchanged, the balance position stays undisturbed. However, the sensitivity of the bridge changes.

In the balanced condition, the resistance in the branch BD could be avoided. 

For example, the resistance connected to BC could be avoided. 

Note - In a Wheatstone bridge, if the battery and the galvanometer are interchanged, the deflection in a galvanometer does not vary.

At the balance point,

When P = Q then ΔR = S α ΔT

Meter Bridge or Slide Wire Bridge

The Meter Bridge is based on the balanced Wheatstone bridge principle. It is used to find unknown resistances. 

Working: Assume P is the unknown resistance.

At the balance point,

Q is known and l can be calculated.

Tips for Students Studying JEE Main Physics

1. A student has to go through the comprehensive syllabus and focus first on the difficult topics and then move on to the easier topics. 
2. In order to develop a deep knowledge of the subject and to crack such a high-level examination, thorough and topic-wise extensive notes are very important. 
3. A student must have the skills of time management because there is a lot of syllabi that a student must study through. 
4. A student needs to find a person who can take up his difficult doubts and become the support mechanism during training. This will also reduce the emotional strain that would usually be found in the students studying for the national level test.