Quick Summary: covering Electric Current (Instantaneous vs Average), Isolated Conductor Behavior at room/zero temperature, Drift Velocity & Microscopic form of Ohm’s Law ($\vec{J} = \sigma\vec{E}$),Current Density, Resistance Variations (Stretching wires, mass/length ratios), Temperature Coefficient of Resistance ($\alpha$), Ohm’s Law Circuit Analysis, Series & Parallel Combinations, Nodal Analysis ($V_M$ calculations for complex loops), and Ideal/Non-Ideal Ammeter & Voltmeter reading techniques.
1. Electric Current Fundamentals & Calculations
A. Definition and Nature of Current
- Rate of directional flow of electric charge with respect to time. It is a scalar quantity (despite having magnitude and direction, it does not follow triangle law of vector addition; measured in Amperes or C/s).
- Direction of current is along the motion of positive charge and opposite to negative charge. Current through a surface does not depend on the area of the surface.
- Instantaneous Current: $$\mathbf{I = \frac{dq}{dt}} \text{ (slope of } q\text{-}t \text{ graph)}$$
- Average Current: $$\mathbf{I_{avg} = \frac{\Delta q}{\Delta t} = \frac{\int I dt}{\int dt}}$$
- Revolving Charge Current: If a charge $q$ revolves in a circular path of radius $R$ with speed $v$ (frequency $f = \frac{v}{2\pi R}$), equivalent current is:
$$\mathbf{I = qf = \frac{qv}{2\pi R}}$$
2. Conductor Microscopic Behaviour & Drift Velocity
A. Isolated Conductor (No Battery Connected)
- Electric field inside ($E_{\text{inside}}$) $= 0$. Free electrons undergo random thermal motion in zig-zag paths at room temperature.
- Average thermal velocity of free electrons $= 0$, but root-mean-square thermal speed is $v_{\text{rms}} \approx 10^5\text{ m/s}$ (proportional to $\sqrt{T}$).
- Relaxation Time ($\tau$): Average time interval between two successive collisions ($\tau \propto \frac{1}{\text{Temperature}}$).
- Free electron density ($n$) for metals $\approx 10^{29}\text{ electrons/m}^3$. Net current ($I$) $= 0$.
B. Conductor Connected to Battery & Drift Velocity
- Electric field established inside: $E = \frac{V}{l}$. Force on electron $F = eE$, acceleration $a = \frac{eE}{m}$.
- Drift Velocity ($v_d$): Average uniform velocity with which free electrons drift towards positive terminal:
$$\mathbf{v_d = \frac{eE\tau}{m} = \frac{eVI}{ml} \quad (\text{ magnitude } \approx 10^{-4}\text{ m/s})}$$ - Relation between Current and Drift Velocity:
$$\mathbf{I = neAv_d}$$
3. Microscopic Ohm’s Law & Resistance Variations
A. Current Density & Vector Ohm’s Law
- Current Density ($\vec{J}$): Current flowing per unit normal area ($J = \frac{I}{A} = nev_d = \sigma E$). Vector form: $\mathbf{\vec{J} = \sigma\vec{E}}$ or $\mathbf{\rho\vec{J} = \vec{E}}$.
- Resistivity ($\rho$) & Conductivity ($\sigma$):
$$\rho = \frac{m}{ne^2\tau}, \quad \sigma = \frac{1}{\rho} = \frac{ne^2\tau}{m}$$ - Mobility ($\mu$): Drift velocity per unit electric field:
$$\mathbf{\mu = \frac{v_d}{E} = \frac{e\tau}{m}}$$
B. Resistance Variations (Stretching Wires)
- Resistance formula: $R = \rho \frac{l}{A}$.
- If length is changed while Volume remains constant ($V = \text{constant}$):
• $R \propto l^2 \implies \frac{\Delta R}{R} \times 100 = 2\left(\frac{\Delta l}{l} \times 100\right)$ (for small changes).
• $R \propto \frac{1}{A^2} \propto \frac{1}{r^4}$.
- If mass ($m$) and length ($l$) ratios are given: $\mathbf{R \propto \frac{l^2}{m}}$.
4. Temperature Dependence, Non-Uniform Conductors & Ohm’s Law
A. Temperature Dependence of Resistance
| Material Type | Effect of Temperature ($T$) Increase | Temperature Coefficient ($\alpha = \frac{dR}{R dT}$) |
|---|---|---|
| Conductors / Metals | Randomness increases $\implies$ relaxation time $\tau$ decreases $\implies$ Resistance increases ($\rho = \frac{m}{ne^2\tau}$). | Positive ($+\text{ve}$) |
| Semiconductors & Insulators | Charge carrier concentration ($n$) increases drastically $\implies$ Resistance decreases. | Negative ($-\text{ve}$) |
B. Ohm’s Law & Potential Drop
- Potential drop across a resistor: $\Delta V = I \cdot R$. Current always flows from High Potential to Low Potential ($V_1 > V_2$).
- Traversing a resistor in the direction of current gives a potential drop ($-IR$); traversing opposite to current gives a potential rise ($+IR$).
5. Resistor Combinations, Nodal Analysis & Circuit Solving
A. Series and Parallel Combinations
- Series Combination: Same current flows through each resistor.
$$\mathbf{R_{\text{eq}} = R_1 + R_2 + R_3 + \dots + R_n \quad (R_{\text{eq}} = nR \text{ for } n \text{ identical resistors})}$$
Potential divides in direct proportion to resistance ($V \propto R$). - Parallel Combination: Same potential difference across each branch.
$$\mathbf{\frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \dots + \frac{1}{R_n} \quad \left(R_{\text{eq}} = \frac{R}{n} \text{ for } n \text{ identical resistors}\right)}$$
Current divides in inverse proportion to resistance ($I \propto \frac{1}{R}$). For two resistors in parallel: $R_{\text{eq}} = \frac{R_1 R_2}{R_1 + R_2}$.
B. Nodal Analysis (Finding Node Potential $V_M$)
- For any junction node $V_M$ connected to surrounding potentials $V_1, V_2$ via resistances $R_1, R_2$:
$$\mathbf{V_M = \frac{\frac{V_1}{R_1} + \frac{V_2}{R_2} + \dots}{\frac{1}{R_1} + \frac{1}{R_2} + \dots} = \frac{\sum \frac{V}{R}}{\sum \frac{1}{R}}}$$
6. Measuring Instruments: Ammeters and Voltmeters
| Instrument | Ideal Characteristics | Non-Ideal / Real Characteristic & Circuit Handling |
|---|---|---|
| Ammeter ($A$) | Resistance of Ideal Ammeter = Zero ($R = 0$). Connected in series. | If given finite resistance, treat it as a resistor with that specific resistance value in series and calculate current passing through it. |
| Voltmeter ($V$) | Resistance of Ideal Voltmeter = Infinite ($R = \infty$). Connected in parallel. Current drawn $I = 0$. | If given finite resistance, treat it as a parallel branch resistor and calculate potential difference across its terminals. |