Current Electricity revision notes

Electric Current and Conductors

Charges in motion constitute an electric current. While lightning is an example of an unsteady current (carrying tens of thousands of Amperes), steady currents are observed in everyday devices like torches, cell-driven clocks, and domestic appliances (which carry currents on the order of 1 Ampere). At the smallest scale, currents in our nerves are in microamperes ($\mu$A). If a net amount of charge $\Delta Q$ flows across a cross-section of a conductor in time $\Delta t$, the current at time $t$ in the limit of $\Delta t \rightarrow 0$ is defined as $I = \lim_{\Delta t \to 0} \frac{\Delta Q}{\Delta t}$. In bulk matter like metals, some electrons are practically free to move and are not bound to individual nuclei. These free electrons carry current in solid conductors, while positive ions remain fixed in the background. In the ionosphere, free charged particles exist naturally. In electrolytic solutions, both positive and negative charges can move to constitute a current.

In the absence of an electric field, free electrons are in random thermal motion, colliding with fixed ions. Because the directions of their velocities are completely random, the average velocity of all electrons is zero ($\frac{1}{N} \sum v_i = 0$), resulting in no net electric current. [JEE TIP] The thermal speed of electrons at room temperature is relatively high (e.g., $\sim 2 \times 10^2$ m/s for copper atoms), but the average bulk velocity over all electrons is exactly zero. When a steady electric field is applied, it accelerates the electrons, resulting in a continuous steady current if the electric field is continuously maintained by a mechanism like a cell or battery.

Ohm's Law and Resistance

Discovered by G.S. Ohm, this law states that the current $I$ flowing through a conductor is proportional to the potential difference $V$ across its ends, expressed as $V \propto I$ or $V = R I$. The proportionality constant $R$ is the resistance of the conductor.

Derivation of Resistance Geometry ($R = \rho l/A$): Resistance $R$ depends on both the material and dimensions.

• Length ($l$): Imagine two identical rectangular slabs of length $l$ and resistance $R$ placed in series. The current $I$ flows through both, but the total potential difference is $V + V = 2V$. The equivalent resistance is $R_C = 2V/I = 2R$. Thus, doubling length doubles resistance ($R \propto l$).
• Area ($A$): Imagine cutting the slab lengthwise into two parallel half-slabs of area $A/2$. For a given voltage $V$, the current splits evenly to $I/2$ in each half. The resistance of a half-slab is $V/(I/2) = 2(V/I) = 2R$. Thus, halving the area doubles the resistance ($R \propto 1/A$). Combining these gives $R = \rho \frac{l}{A}$, where $\rho$ is the resistivity of the material.

Color Coding of Carbon Resistors: Commercial carbon resistors have their resistance values indicated by a color code. The first two bands indicate the first two significant figures, the third band indicates the decimal multiplier, and the fourth band indicates the tolerance.

• Color Sequence: Black (0), Brown (1), Red (2), Orange (3), Yellow (4), Green (5), Blue (6), Violet (7), Gray (8), White (9).
• Tolerance: Gold ($\pm 5\%$), Silver ($\pm 10\%$), No Color ($\pm 20\%$).JEE TIPMnemonic to remember the color code: B B ROY of Great Britain had a Very Good Wife.

Current Density, Drift Velocity, and Conductivity

Current per unit area taken normal to the current is defined as current density $j = I/A$. If $E$ is the magnitude of a uniform electric field in a conductor of length $l$, the potential difference is $V = El$. Using $V = IR$ and $R = \rho l/A$, we get $E = j\rho$. In vector form, this is $\mathbf{j} = \sigma \mathbf{E}$, where $\sigma \equiv 1/\rho$ is the conductivity.

When an electric field $E$ is present, electrons accelerate with $a = -eE/m$. However, they do not accelerate indefinitely; they undergo collisions with heavy fixed ions at random intervals. The average time between successive collisions is the relaxation time $\tau$. Averaging the velocities of all electrons yields a steady, time-independent average velocity called the drift velocity: $v_d = -eE\tau/m$.

The total charge crossing an area $A$ in time $\Delta t$ is $\Delta Q = neA|v_d|\Delta t$, where $n$ is the number of free electrons per unit volume. For copper, this number density is enormous, $n \approx 8.5 \times 10^{28}$ m$^{-3}$. This relates current and drift velocity: $I = neA|v_d|$. By substituting $v_d$, we derive the microscopic form of Ohm's Law: $j = \left(\frac{ne^2\tau}{m}\right)E$. Thus, conductivity is $\sigma = \frac{ne^2\tau}{m}$ and resistivity is $\rho = \frac{m}{ne^2\tau}$.

Another important quantity is mobility $\mu$, defined as the magnitude of drift velocity per unit electric field: $\mu = \frac{|v_d|}{E} = \frac{e\tau}{m}$. Mobility is inherently always positive.

Temperature Dependence of Resistivity

Materials are classified by resistivity into metals ($10^{-8}$ to $10^{-6} \Omega$m), insulators ($10^{22}$ to $10^{24}$ times greater), and semiconductors (intermediate). Over a limited temperature range, the resistivity of a metallic conductor is approximately $\rho_T = \rho_0 [1 + \alpha (T-T_0)]$, where $\alpha$ is the temperature co-efficient of resistivity. For metals, $\alpha$ is positive. As temperature increases, the average thermal speed of electrons increases, causing more frequent collisions, which decreases the relaxation time $\tau$. Since $n$ is roughly constant in metals, a decrease in $\tau$ causes resistivity $\rho$ to increase. For insulators and semiconductors, the number density of free charge carriers $n$ increases with temperature. This exponential increase in $n$ more than compensates for any decrease in $\tau$, so their overall resistivity decreases rapidly with temperature ($\alpha$ is negative).

Electrical Energy and Power

When charge $\Delta Q$ moves through a potential difference $V$ across a conductor from A to B, its potential energy changes by $\Delta U_{pot} = -IV\Delta t$. If charges moved without collisions, their kinetic energy would increase. However, due to collisions with ions, charges move with a steady drift velocity, and the energy gained is shared with vibrating atoms, causing the conductor to heat up. The power dissipated (energy per unit time) is $P = IV$. Using Ohm's law, this ohmic loss can be written as $P = I^2 R = \frac{V^2}{R}$. To transmit power $P$ over cables with resistance $R_c$, the power wasted as heat is $P_c = I^2 R_c = \frac{P^2 R_c}{V^2}$.

Maximum Power Transfer Theorem: In a circuit where a cell of EMF $\epsilon$ and internal resistance $r$ supplies current to an external variable resistor $R$, the power dissipated in the external resistor is $P = \left(\frac{\epsilon}{R+r}\right)^2 R$. [JEE TIP] The power delivered to the external circuit is maximized strictly when the external resistance equals the internal resistance ($R = r$). The maximum power delivered is $P_{max} = \frac{\epsilon^2}{4r}$.

Cells, EMF, Internal Resistance, and Combinations

An electrolytic cell consists of positive (P) and negative (N) electrodes immersed in an electrolyte. Dipped in the solution, the electrodes exchange charges. The positive electrode develops a potential $V_+ > 0$ relative to the adjacent electrolyte. The negative electrode develops a potential $-V_- \le 0$ relative to the adjacent electrolyte. When no current flows (open circuit), the potential difference across the whole cell is the electromotive force (emf) $\epsilon = V_+ + V_- > 0$.

When a current $I$ is drawn, the electrolyte offers an internal resistance $r$. Current flows from N to P inside the electrolyte. The terminal voltage across the cell drops to $V = \epsilon - Ir$ (while discharging). The maximum possible current from a cell is $I_{max} = \epsilon/r$.

Combinations of Cells:

1. Series Combination (n cells): When the negative terminal of one cell is connected to the positive of another: $\epsilon_{eq} = \epsilon_1 + \epsilon_2 + \dots + \epsilon_n$ and $r_{eq} = r_1 + r_2 + \dots + r_n$.
2. Parallel Combination (n cells): $\frac{\epsilon_{eq}}{r_{eq}} = \frac{\epsilon_1}{r_1} + \dots + \frac{\epsilon_n}{r_n}$ and $\frac{1}{r_{eq}} = \frac{1}{r_1} + \dots + \frac{1}{r_n}$.
3. Mixed Grouping of Cells: For a grid of cells arranged in $m$ parallel rows, with each row having $n$ cells in series (total cells = $mn$), the equivalent EMF is $n\epsilon$ and the equivalent internal resistance is $\frac{nr}{m}$. The current drawn by external resistance $R$ is $I = \frac{mn\epsilon}{mR + nr}$.JEE TIPMaximum current in mixed grouping is obtained when the external resistance equals the total equivalent internal resistance of the network ($R = \frac{nr}{m}$).

Kirchhoff's Rules, Complex Circuits, and Measuring Instruments

Complex circuits are analyzed using Kirchhoff's rules:

1. Junction Rule: At any junction, the sum of the currents entering the junction is equal to the sum of currents leaving the junction. This is a direct consequence of the conservation of electric charge.
2. Loop Rule: The algebraic sum of changes in potential around any closed loop involving resistors and cells is zero. This is based on the conservation of energy.

Wheatstone Bridge & Meter Bridge: A Wheatstone bridge has four resistors $R_1, R_2, R_3, R_4$, a battery across one diagonal, and a galvanometer across the other. The bridge is balanced when the current through the galvanometer $I_g = 0$. Applying Kirchhoff's rules yields the balance condition: $\frac{R_2}{R_1} = \frac{R_4}{R_3}$. A Meter Bridge is the practical realization of this, where a 1-meter wire acts as two ratio arms $l$ and $(100-l)$. The unknown resistance is found by $R = S \frac{l}{100-l}$.

Potentiometer: A versatile instrument used for measuring potential differences, comparing EMFs, and finding internal resistance without drawing current from the cell under test.

• Principle: For a uniform wire carrying steady current, the potential drop across any length is directly proportional to that length ($V = kl$, where $k$ is the potential gradient).
• Comparing EMFs: $\frac{\epsilon_1}{\epsilon_2} = \frac{l_1}{l_2}$ where $l_1, l_2$ are balancing lengths.
• Finding Internal Resistance: $r = R\left(\frac{l_1}{l_2} - 1\right)$, where $l_1$ is the open-circuit balance length and $l_2$ is the balance length when cell is shunted with resistor $R$.
• JEE TIPThe potentiometer acts as an ideal voltmeter because it draws exactly zero current from the test circuit at the null point.

Galvanometer Conversion:

• Ammeter: A galvanometer is converted to an ammeter by connecting a very small resistance (shunt, $S$) in parallel. $S = \frac{I_g G}{I - I_g}$.
• Voltmeter: A galvanometer is converted to a voltmeter by connecting a very large resistance ($R$) in series. $R = \frac{V}{I_g} - G$.

Advanced Circuit Techniques

[JEE TIP] Symmetry Techniques for Complex Grids:

1. Mirror Symmetry: If a circuit is symmetric about an axis perpendicular to the line joining the input and output nodes, nodes that are mirror images of each other are at identical potentials. You can safely split or disconnect joints at the axis of symmetry.
2. Folding Symmetry: If a circuit is symmetric about the axis strictly parallel to the direction of current flow, points mapped onto each other by folding have the same potential.
3. Infinite Ladder Networks: To find the equivalent resistance $R_{eq}$ of a ladder stretching to infinity, assume the entire resistance is $R_{eq}$. By adding one additional repeating unit to the front of $R_{eq}$, the total resistance remains identically $R_{eq}$. Set up the quadratic equation and solve.

Key Concepts & Definitions

Electric Current

The net amount of positive charge flowing forward minus backward across an area per unit time. 1 Ampere is formally defined through the magnetic effects of currents.

Relaxation Time (τ\tauτ)

The average time elapsed between successive collisions of an electron with the heavy fixed ions in a conductor.

Current Density (j\mathbf{j}j)

The current flowing per unit area taken normal to the current. It is a vector directed along the electric field.

Conductivity (σ\sigmaσ)

The reciprocal of resistivity (σ=1/ρ\sigma = 1/\rhoσ=1/ρ), relating current density to electric field.

Mobility (μ\muμ)

The magnitude of the drift velocity acquired per unit applied electric field.

Electromotive Force (EMF)

The potential difference between the positive and negative electrodes of a cell in an open circuit.

Conditions & Limitations

• Validity of Ohm's Law: Ohm's law is not a fundamental law of nature. It fails when:
  1. $V$ ceases to be proportional to $I$ non-linearly (e.g., when a good conductor heats up).
  2. The relation depends on the sign of $V$ (e.g., in a diode, reversing voltage does not produce the same current magnitude).
  3. The relation is non-unique, where multiple $V$ values correspond to the same $I$ (e.g., GaAs).
• High Electric Field Limit: Even for homogeneous conductors like pure silver or semiconductors, Ohm's law is only obeyed within a specific range of electric fields. If the electric field becomes too strong, there are extreme departures from Ohm's law in all cases.
• Temperature Dependence Formula: The linear approximation $\rho_T = \rho_0 [1 + \alpha(T-T_0)]$ is valid only over a limited range of temperatures (it fails at temperatures much lower than $0^\circ$C).

Important Graphs & Diagrams

• Non-Ohmic V-I Graphs:
  • Good Conductor Heating Up: At higher currents, the curve deviates upward from the straight dashed line because resistance increases with temperature.
  • Diode: Asymmetrical curve where a positive voltage yields significant current (mA scale) and negative voltage yields almost zero current until breakdown ($\mu$A scale).
  • GaAs: Exhibits a non-unique relationship where the curve rises to a peak and then dips, showing a negative resistance region where multiple voltage values produce the same current.
• Resistivity vs. Temperature ($\rho-T$) Graphs:
  • Copper: Curve bends non-linearly at very low temperatures (below $0^\circ$C) but becomes a straight line at higher temperatures.
  • Nichrome: A straight line with a very small slope, intercepting the y-axis at a very high initial resistivity value.JEE TIPAlloys like Nichrome, Manganin, and Constantan exhibit a very weak (almost flat) dependence of resistivity on temperature, making them ideal for standard resistors.
  • Semiconductor: A non-linear decay curve showing resistivity dropping as absolute temperature increases.

Standard Derivations & Step-by-Step Problem Solving

Applying Kirchhoff’s Rules to Complex Networks (e.g., Wheatstone Bridge Balance):

1. Assign unknown currents to each branch. Use the Junction Rule first to minimize unknowns by splitting currents logically at nodes (e.g., $I_1 = I_3$, $I_2 = I_4$ if $I_g = 0$).
2. Choose closed loops and define a traversal direction.
3. Apply the Loop Rule ($\sum \Delta V = 0$). Drops in potential across resistors in the direction of current are taken as negative ($-IR$). Cell emfs are positive if traversed from negative to positive terminals.
4. For a balanced bridge with $I_g = 0$:
   • Loop 1 (ADBA): $-I_1R_1 + 0 + I_2R_2 = 0 \Rightarrow I_1R_1 = I_2R_2$.
   • Loop 2 (CBDC): $-I_3R_3 + 0 + I_4R_4 = 0$. Since $I_1=I_3$ and $I_2=I_4$, this becomes $-I_1R_3 + I_2R_4 = 0 \Rightarrow I_1R_3 = I_2R_4$.
   • Divide the two equations: $\frac{R_2}{R_1} = \frac{R_4}{R_3}$.

Solving Infinite Ladder Networks:

1. Let the equivalent resistance of the entire infinite network be $R_{eq}$.
2. Identify the fundamental repeating unit of the ladder.
3. Because the ladder is infinite, adding one more repeating unit to the front of the ladder does not change the total resistance $R_{eq}$.
4. Construct the circuit equation with the first unit attached to $R_{eq}$, set it equal to $R_{eq}$, and solve the resulting quadratic equation. Reject the negative resistance root.

Previous Year JEE Topics

• Wheatstone Bridge & Meter Bridge: Utilizing the null condition ($R_2/R_1 = R_4/R_3$) to find unknown resistances, and analyzing the impact of galvanometer/battery position swapping (it doesn't affect the null condition).
• Potentiometer Mechanics: Calculating internal resistance of cells and comparing non-standard EMFs using $r = R(l_1/l_2 - 1)$.
• Color Coding & Tolerance: Reading equivalent resistance values directly from color bands (e.g., Red Red Orange = $22 \times 10^3 \Omega$).
• Resistor and Cell Combinations: Converting complex networks using series/parallel simplifications, infinite ladders, and identifying symmetry (as in the cubical network where equivalent resistance across a main diagonal is $\frac{5}{6}R$, face diagonal is $\frac{3}{4}R$, edge is $\frac{7}{12}R$).
• Temperature Coefficient of Resistance: Using $\rho_T = \rho_0 [1 + \alpha(T-T_0)]$ to find operating temperatures of heating elements (e.g., toaster elements settling at steady resistance due to heat balance).
• Drift Velocity & Microscopic Ohm's Law: Relationships between drift velocity, current density, and relaxation time ($j = ne^2\tau E / m$).
• Power and Energy: Calculating power lost in transmission lines, efficiency $P_c = P^2 R_c/V^2$, and applying the Maximum Power Transfer Theorem.