Prerequisites
Before studying this topic, make sure you understand:
- Drift Velocity - How electrons move in conductors
- Electric Potential - Voltage fundamentals
The Hook: Why Don’t All Wires Glow Like Bulbs?
Ever noticed how your phone charger cable stays cool while charging, but a small LED bulb glows bright with the same voltage? Both have current flowing through them, so why the difference?
The answer: Different materials have different resistance to current flow. This simple relationship between voltage, current, and resistance - discovered by Georg Ohm in 1827 - powers every circuit in the world, from Iron Man’s arc reactor to your smartphone!
Challenge question: If you double the voltage across a wire, what happens to the current? Keep reading to find out!
Interactive Demo
Adjust voltage and see how current changes for different materials:
Formula Visualization: Understanding V = IR
See what each variable represents in a real circuit:
The Core Concept: Ohm’s Law
The Fundamental Relationship
$$\boxed{V = IR}$$In simple terms: “Voltage is the push, Current is the flow, Resistance is the opposition”
Alternative forms:
- $I = \frac{V}{R}$ (current in terms of voltage)
- $R = \frac{V}{I}$ (resistance from V-I measurement)
Think of electricity like water flowing through pipes:
| Electrical | Water System |
|---|---|
| Voltage (V) | Water pressure |
| Current (I) | Water flow rate |
| Resistance (R) | Pipe narrowness/friction |
Higher pressure (V) → More water flow (I) Narrower pipe (higher R) → Less water flow (I)
This is exactly how Ohm’s law works!
Units
- Voltage (V): Volt (V) = Joule/Coulomb
- Current (I): Ampere (A) = Coulomb/second
- Resistance (R): Ohm (Ω) = Volt/Ampere
Ohmic vs Non-Ohmic Conductors
Ohmic Conductors
Definition: Materials that obey Ohm’s law - current is directly proportional to voltage.
$$I \propto V \quad \text{(at constant temperature)}$$V-I Graph: Straight line through origin
I ↑
| /
| /
| / (slope = 1/R)
| /
| /
|/________→ V
Examples:
- Metallic conductors (copper, aluminum, silver)
- Carbon resistors
- Nichrome wire (used in heaters)
Key property: Resistance $R$ remains constant (at constant temperature)
Non-Ohmic Conductors
Definition: Materials where current is NOT directly proportional to voltage.
V-I Graph: NOT a straight line
Examples:
Diode (Semiconductor)
- Current flows easily in one direction
- High resistance in reverse direction
- Exponential I-V relationship
Filament Bulb
- Resistance increases as it heats up
- I-V curve bends (current increases slower than voltage)
LED (Light Emitting Diode)
- No current until threshold voltage
- Then sudden current increase
Electrolyte solutions
- I-V characteristic depends on ion concentration
Wrong thinking: “Ohm’s law applies to all materials”
Reality: Ohm’s law is NOT a universal law! It only works for ohmic conductors at constant temperature.
JEE Trap: Questions may give a non-linear V-I graph and ask if Ohm’s law applies - answer is NO!
V-I Characteristics Graphs
1. Metallic Conductor (Ohmic)
I ↑
| /
| / Straight line
| / R = constant
| /
| /
|/________→ V
Slope = 1/R
- Linear relationship
- Passes through origin
- Constant resistance
2. Tungsten Filament Bulb
I ↑
| ___---
| __/ Curve bends
| / (R increases with I)
| /
|/________→ V
Why it bends:
- Current heats the filament
- Temperature ↑ → Resistance ↑
- So current increases slower than voltage
3. Diode (Semiconductor)
I ↑ Forward bias
| |
| |___
| |
|________|_______→ V
| Reverse
| bias
Key features:
- Threshold voltage needed in forward bias
- Very little current in reverse bias
- Used as one-way valve for current
4. Electrolyte
I ↑
| /
| / Initially curved
| / (then becomes linear)
| /
|/________→ V
Behavior:
- Non-linear at low voltages
- Becomes approximately ohmic at higher voltages
Limitations of Ohm’s Law
Temperature changes:
- Resistance varies with temperature
- Must specify “at constant temperature”
Non-ohmic materials:
- Semiconductors (diodes, transistors)
- Vacuum tubes
- Electrolytes (at low voltages)
Very high electric fields:
- Breakdown voltage exceeded
- Material properties change
AC circuits with inductors/capacitors:
- Need to use impedance, not resistance
- Phase difference between V and I
Microscopic Form of Ohm’s Law
From drift velocity concepts:
$$J = \sigma E$$where:
- $J$ = current density
- $\sigma$ = conductivity
- $E$ = electric field
Derivation connection:
$$I = nAev_d$$ $$v_d = \mu E = \frac{e\tau}{m}E$$ $$\therefore I = nAe \cdot \frac{e\tau}{m}E$$With $E = \frac{V}{L}$ and $J = \frac{I}{A}$:
$$J = \sigma E \quad \text{where } \sigma = \frac{ne^2\tau}{m}$$This is Ohm’s law in vector form!
Memory Tricks & Patterns
Mnemonic for Ohm’s Law
“Very Interesting Results” → $V = I \times R$
Or think: “Voltage Is Really important” → $V = IR$
Remembering Forms
Triangle method:
V
___
| | |
|I|R|
|_|_|
- Cover V → $V = I \times R$
- Cover I → $I = V / R$
- Cover R → $R = V / I$
Graph Recognition Patterns
| Graph Shape | Material Type | Ohm’s Law? |
|---|---|---|
| Straight line through origin | Metallic conductor | YES |
| Curved upward (bending) | Filament bulb | NO |
| Steep rise after threshold | Diode/LED | NO |
| Flat then rising | Electrolyte | NO (initially) |
JEE Exam Trick
If graph is straight: $R = \frac{V}{I} = \frac{1}{\text{slope}}$
If graph is curved: $R = \frac{V}{I}$ at that POINT (not constant!)
When to Use This
Given two, find third:
- Know V and R → Find I = V/R
- Know I and R → Find V = IR
- Know V and I → Find R = V/I
For V-I graph questions:
- Straight line → Ohmic → Use $R = 1/\text{slope}$
- Curved → Non-ohmic → Resistance varies with V or I
Word problem clues:
- “Metallic wire” → Assume ohmic
- “Bulb”, “Diode”, “LED” → Non-ohmic
- “At constant temperature” → Confirms ohmic behavior
Common Mistakes to Avoid
Wrong: “This question has V and I, so V = IR applies”
Correct: First check if material is ohmic!
- Metals at constant T → Ohmic ✓
- Semiconductors, bulbs, electrolytes → Non-ohmic ✗
JEE Insight: If graph is given, it’s probably testing your understanding of NON-ohmic behavior!
Wrong: “Slope of V-I graph = Resistance”
Correct:
- Slope of V-I graph = $\frac{dV}{dI} = R$
- Slope of I-V graph = $\frac{dI}{dV} = \frac{1}{R}$ = Conductance
Memory trick: More common is I-V graph, where slope = 1/R
Which axis is which?
- V-I graph: V on y-axis, I on x-axis → slope = R
- I-V graph: I on y-axis, V on x-axis → slope = 1/R
Wrong: “Wire resistance is always constant”
Correct: Resistance changes with temperature!
- For metals: $R$ increases with T (positive temperature coefficient)
- For semiconductors: $R$ decreases with T (negative temperature coefficient)
JEE application: Filament bulb problems - as current increases, bulb heats up, R increases, so I-V curve bends!
Wrong: “Resistance can be negative if current flows backward”
Correct:
- Resistance is ALWAYS positive (for passive elements)
- Current direction determines sign of $I$, not $R$
- If you get negative R, you made a sign error!
Exception: Active devices like tunnel diodes can show negative differential resistance in specific regions (but that’s beyond JEE scope).
Practice Problems
Level 1: Foundation (NCERT/Basic)
A wire carries a current of 2 A when connected to a 12 V battery. Find its resistance.
Solution:
Using Ohm’s law: $V = IR$
$$R = \frac{V}{I} = \frac{12}{2} = 6 \text{ Ω}$$Answer: $R = 6$ Ω
Concept check: This assumes the wire is ohmic (metallic conductor).
A resistor of 100 Ω is connected across a 220 V supply. Calculate the current through it.
Solution:
$$I = \frac{V}{R} = \frac{220}{100} = 2.2 \text{ A}$$Answer: $I = 2.2$ A
Safety note: This is a significant current - the resistor will heat up! (We’ll study this in electrical power).
Which of the following obey Ohm’s law? (a) Copper wire (b) LED (c) Diode (d) Nichrome wire
Solution:
Ohmic conductors: (a) Copper wire and (d) Nichrome wire
- Both are metallic conductors
- Show linear V-I characteristics at constant temperature
Non-ohmic: (b) LED and (c) Diode
- Semiconductor devices
- Non-linear V-I characteristics
Answer: (a) and (d) obey Ohm’s law
Level 2: JEE Main
The V-I graph for a conductor makes an angle of 30° with the V-axis. Find the resistance of the conductor.
Solution:
In V-I graph (V on y-axis, I on x-axis):
$$\text{slope} = \tan(30°) = \frac{V}{I} = R$$ $$R = \tan(30°) = \frac{1}{\sqrt{3}} = 0.577 \text{ Ω}$$Answer: $R = \frac{1}{\sqrt{3}}$ Ω
Key insight: Angle with V-axis gives you the resistance directly!
A 220 V, 100 W bulb is connected to a 110 V supply. What is the power consumed? (Assume resistance remains constant)
Solution:
First find resistance of bulb:
$$P = \frac{V^2}{R}$$ $$R = \frac{V^2}{P} = \frac{(220)^2}{100} = 484 \text{ Ω}$$At 110 V:
$$P' = \frac{V'^2}{R} = \frac{(110)^2}{484} = \frac{12100}{484} = 25 \text{ W}$$Answer: $P' = 25$ W
Key insight: Power ∝ V² for constant R, so halving voltage reduces power to 1/4!
Reality check: This assumes resistance stays constant, but for a real bulb, resistance decreases when cooler, so actual power would be slightly higher than 25 W.
A battery of 12 V is connected to a resistor. The ammeter shows 3 A. If the voltmeter is connected across the resistor, what will it read?
Solution:
This tests understanding of circuit measurement!
Ammeter reading: $I = 3$ A (measures current through resistor)
By Ohm’s law:
$$R = \frac{V}{I}$$But we need V! Since battery is 12 V and no internal resistance mentioned:
Voltmeter reading = Voltage across resistor = 12 V
(Assuming ideal battery with no internal resistance)
Using this:
$$R = \frac{12}{3} = 4 \text{ Ω}$$Answer: Voltmeter reads 12 V
Circuit concept: Voltmeter measures potential difference across the resistor, which equals battery voltage (for ideal battery).
Level 3: JEE Advanced
The I-V characteristic of a non-ohmic conductor is given by $I = 2V + 3V^2$ (where I is in A and V is in volts). Find the resistance when V = 1 V and V = 2 V.
Solution:
For non-ohmic conductor, resistance varies with voltage.
At V = 1 V:
$$I = 2(1) + 3(1)^2 = 2 + 3 = 5 \text{ A}$$ $$R = \frac{V}{I} = \frac{1}{5} = 0.2 \text{ Ω}$$At V = 2 V:
$$I = 2(2) + 3(2)^2 = 4 + 12 = 16 \text{ A}$$ $$R = \frac{V}{I} = \frac{2}{16} = 0.125 \text{ Ω}$$Answer:
- At V = 1 V: R = 0.2 Ω
- At V = 2 V: R = 0.125 Ω
Advanced insight: Resistance DECREASES as voltage increases for this material. This is characteristic of certain semiconductors and electrolytes!
Differential resistance:
$$r = \frac{dV}{dI} = \frac{1}{\frac{dI}{dV}} = \frac{1}{2 + 6V}$$At V = 1 V: $r = 1/8 = 0.125$ Ω (dynamic resistance)
Two wires of the same material have lengths in the ratio 1:2 and radii in the ratio 2:1. They are connected (a) in series, (b) in parallel across a battery. Find the ratio of currents through them in both cases.
Solution:
Resistance: $R = \rho \frac{L}{A} = \rho \frac{L}{\pi r^2}$
For wire 1: $R_1 = \rho \frac{L_1}{\pi r_1^2}$
For wire 2: $R_2 = \rho \frac{L_2}{\pi r_2^2}$
$$\frac{R_2}{R_1} = \frac{L_2}{L_1} \times \frac{r_1^2}{r_2^2} = \frac{2}{1} \times \frac{(2)^2}{(1)^2} = 2 \times 4 = 8$$So $R_2 = 8R_1$
(a) In Series: Same current through both wires!
$$\frac{I_1}{I_2} = \frac{1}{1} = 1:1$$(b) In Parallel: Same voltage across both wires.
$$I = \frac{V}{R}$$ $$\frac{I_1}{I_2} = \frac{R_2}{R_1} = \frac{8}{1} = 8:1$$Answer:
- Series: $I_1 : I_2 = 1:1$
- Parallel: $I_1 : I_2 = 8:1$
Key concepts:
- Series → Same current
- Parallel → Current inversely proportional to resistance
The I-V characteristic of a material is shown below. Identify the region where the material obeys Ohm’s law, and calculate the resistance in that region.
(Assume graph shows: Linear from 0-2V, then curved from 2-5V)
Solution:
Ohmic region: 0 to 2 V (linear portion)
Let’s say at V = 2 V, I = 0.5 A (from graph)
In linear region:
$$R = \frac{V}{I} = \frac{2}{0.5} = 4 \text{ Ω}$$Beyond 2 V: Material becomes non-ohmic
- Resistance is not constant
- Must calculate $R = V/I$ at each point
Answer:
- Ohmic region: 0-2 V
- Resistance in ohmic region: 4 Ω
- Beyond 2 V: Non-ohmic behavior
JEE Strategy: Always identify the LINEAR portion of V-I graphs - that’s where Ohm’s law applies!
Quick Revision Box
| Situation | Formula/Approach |
|---|---|
| Basic Ohm’s law | $V = IR$ |
| Finding resistance from V-I graph (linear) | $R = \frac{V}{I} = \frac{1}{\text{slope of I-V graph}}$ |
| Metallic conductor at constant T | Ohmic (linear V-I) |
| Filament bulb, diode, LED | Non-ohmic (curved V-I) |
| Resistance from I-V curve | $R = V/I$ at that point |
| Microscopic form | $J = \sigma E$ |
| Power from Ohm’s law | $P = VI = I^2R = V^2/R$ |
JEE Strategy: High-Yield Points
Graph interpretation
- Identifying ohmic vs non-ohmic from V-I curves
- Calculating resistance from slope
- Appears in 2-3 questions every year!
Conceptual understanding
- Why Ohm’s law fails for certain materials
- Temperature effects on resistance
- Often asked as assertion-reason questions
Numerical problems
- Quick calculations with $V = IR$
- Combined with power formulas
- Multi-step problems with series/parallel circuits
Common traps
- Confusing slope of V-I vs I-V graphs
- Assuming all materials are ohmic
- Forgetting temperature dependence
Time-saving tricks:
- Recognize graph shapes instantly (practice 10-15 graphs)
- Use triangle method for quick formula recall
- For non-ohmic: Calculate R at specific points, don’t assume constant
Real-World Applications
Phone chargers
- Converts 220V AC to 5V DC
- Resistance of charging cable determines current
Filament bulbs (Jawan, Oppenheimer scenes)
- Non-ohmic behavior due to heating
- That’s why bulbs often fail when switched ON (cold → low R → high surge current)
LED lights
- Don’t follow Ohm’s law
- Need current-limiting resistors
Electric heaters (Nichrome wire)
- Ohmic at steady state
- Constant resistance = predictable heating
Iron Man’s arc reactor
- Superconducting components (zero resistance!)
- Normal conductors would waste too much energy as heat
Related Topics
Within Current Electricity
- Drift Velocity - Microscopic origin of Ohm’s law
- Resistance and Resistivity - Material properties affecting V-I characteristics
- Electrical Power - Energy dissipation in resistors
- Kirchhoff’s Laws - Applying Ohm’s law to complex circuits
Connected Chapters
- Electrostatics - Voltage as potential difference
- Magnetism - V-I characteristics of inductors
- Semiconductors - Non-ohmic behavior explained
Math Connections
- Linear Equations - Straight-line V-I graphs
- Graphs and Functions - Interpreting V-I characteristics
- Differentiation - Dynamic resistance = dV/dI
Teacher’s Summary
Ohm’s law $V = IR$ relates voltage, current, and resistance - but only for ohmic conductors at constant temperature!
Ohmic conductors show linear V-I graphs - metallic wires, resistors, nichrome. Non-ohmic show curved graphs - diodes, LEDs, bulbs, electrolytes.
Resistance from graphs: For I-V graph (most common), slope = 1/R. For V-I graph, slope = R.
Temperature matters! Metals increase resistance when heated (filament bulb curve bends). Semiconductors decrease resistance when heated.
The microscopic form $J = \sigma E$ is Ohm’s law in vector form - connects to drift velocity and material properties.
“Ohm’s law is like a straight road - only some materials follow it. Others take curved paths. Know who’s who, and you’ll ace JEE questions!”