Current Electricity | Class 12 CBSE Physics | Concept Notes
Ohm’s law is a special case, not a universal law
It’s easy to come away from Ohm’s law thinking every conductor obeys V = IR with a fixed R. It doesn’t. Ohm’s law is an empirical rule that happens to hold well for metallic conductors at constant temperature — it is not a fundamental law of nature the way Kirchhoff’s laws are. Plenty of everyday components — filament bulbs, diodes, transistors — deliberately violate it, and recognizing how they violate it is exactly what a V-I graph is for.
Reading a V-I graph properly
For a component, plot the potential difference V across it on the y-axis against the current I through it on the x-axis.
The shape of the V–I graph tells you how the component responds to changes in current.
What does the graph shape tell you?
| Graph shape | What it means |
| Straight line through the origin | Ohmic behaviour: V is proportional to I. The resistance is constant, R = V/I |
| Straight line, but not through the origin | The component has a linear V–I relationship with an offset, but it is not ohmic in the strict sense because V/I |
| Smooth curve | Non-ohmic behaviour: the resistance changes with the operating point. The ratio R = V/I |
| Curve with changing slope | The changing slope means that the dynamic resistance changes as the operating point changes. The component therefore cannot be represented by one constant resistance over the entire graph. |
| Loop or multiple-valued V–I curve | The component may exhibit hysteresis: its behaviour depends not only on the instantaneous V or I, but also on its previous state or history. |
Static and dynamic resistance
For a point (I, V) on a V–I graph:
Static resistance
R = V/I
This is the ratio of the voltage to the current at that particular operating point. Graphically, it represents the slope of the line joining the origin to that point.
Dynamic resistance
r = dV/dI
This is the local slope of the tangent to the V–I curve at that operating point.
For a perfectly ohmic resistor:
R = V/I = dV/dI= constant
For a non-ohmic component, V/Iand dV/dI are generally different.
The single biggest mistake students make
Do not calculate R = V/I at one point on a curved V–I graph and then assume that the same resistance applies at a very different voltage.
For a non-ohmic device, resistance depends on the operating point.
Instead, always ask:
“Which resistance am I being asked for — V/I or the local slope dV/dI?”
Quick exam rule
V–I graph:
- Straight line through origin → Ohmic
- Straight line not through origin → Linear but non-ohmic
- Curve → non-ohmic
- Tangent slope → Dynamic resistance, dV/dI
- V/I
→ Static resistance
- Loop → Possible hysteresis/history-dependent behaviour
Important slope reminder
Because the graph has V on the y-axis and I on the x-axis:
Slope = dV/dI
Therefore, for an ohmic resistor:
Slope = R
But if you plot I against V instead, the slope becomes:
Slope = ΔI/ΔV = 1/R
So always check which quantity is on each axis before interpreting the slope.
Exam Tip
A very common question gives a curved V–I graph and asks for the resistance at a particular point.
If the question asks for V/I, use the coordinates of that point.
If it asks for dynamic or differential resistance, find the slope of the tangent at that point.
These two answers need not be the same for a non-ohmic component.
The single biggest mistake students make: computing R = V/I at one point on a curved graph and then using that same R to predict I at a very different V. On a non-ohmic device, R itself is a function of the operating point — you cannot extrapolate a single R value across the graph.
A Comprehensive Guide to V-I Graphs and Resistance Concepts
Case study 1 — the filament (incandescent) bulb
Why Does a Tungsten Filament Bulb Have a Curved V–I Graph?
A tungsten filament bulb has a curved V–I graph because the resistance of its filament changes significantly as its temperature changes.
As current flows through the filament, electrical energy is converted into heat. This Joule heating raises the temperature of the tungsten filament.
Tungsten has a positive temperature coefficient of resistance: its resistance increases as its temperature increases.
Therefore:
More current → more heating → higher temperature → higher resistance
The increasing resistance means that progressively larger increases in voltage are required to produce the same increase in current.
What happens at low voltage?
At low voltage, the filament is relatively cool.
- Temperature is low
- Resistance is relatively low
- A small increase in voltage produces a relatively large increase in current
- Therefore, the V–I graph is relatively flat
- The slope, dV/dI, is relatively small
What happens at high voltage?
At higher voltage, the filament becomes much hotter.
- Temperature increases
- Resistance increases substantially
- The same increase in voltage produces a smaller increase in current
- Therefore, the V–I graph becomes steeper
- The slope, , dV/dI, increases
In other words:
Cool filament → low resistance → flatter V–I curve
Hot filament → high resistance → steeper V–I curve
Why exactly does the graph curve?
For an ohmic resistor, resistance remains constant, so:
V = IR
and the V–I graph is a straight line through the origin.
For a tungsten filament, however, the resistance is not constant:
R increases as filament temperature increases.
Consequently, the relationship between V and I is no longer proportional, and the V–I graph becomes curved.
Static and dynamic resistance
At any particular operating point, the static resistance is:
R = V/I
The dynamic (differential) resistance is given by the local slope:
r = dV/dI
For the tungsten filament, both the operating-point resistance and the local slope change as the filament temperature changes.
Because the V–I curve becomes progressively steeper:
dV/dI increases as the filament gets hotter.
Important clarification
The filament’s resistance does not increase simply because “more current is flowing” as an isolated effect.
The chain of cause and effect is:
Current → Joule heating → temperature rise → increased tungsten resistance
This temperature-dependent resistance is what produces the non-ohmic behaviour.
Is the tungsten filament intrinsically non-ohmic?
The filament is non-ohmic under normal operating conditions because its temperature changes significantly with the applied electrical power.
However, if the filament could be maintained at a constant temperature, its resistance would be approximately constant over a sufficiently small operating range, and its V–I relationship would be approximately linear.
This is why the bulb is a classic example of thermal non-ohmic behaviour.
Energy and power connection
The heating is associated with electrical power:
P = VI
and, equivalently,
P = I²R = V²/R
As the electrical power increases, the filament heats up, causing its resistance to rise.
This creates important feedback:
More electrical power → hotter filament → higher resistance → less increase in current for a given increase in voltage
Exam-ready summary
A tungsten filament bulb shows a curved V–I characteristic because the filament heats up when current flows through it. Tungsten has a positive temperature coefficient of resistance, so its resistance increases with temperature. At low voltage the filament is cool and has relatively low resistance, giving a flatter V–I curve. At higher voltage the filament becomes hotter and its resistance increases, so the V–I curve becomes progressively steeper.
Key idea:
Thermal heating → temperature rise → resistance increase → curved V–I characteristic
This is called thermal non-ohmic behaviour.
Common student mistake
Do not say:
“The resistance increases because the current increases.”
A more accurate explanation is:
“As current increases, Joule heating raises the filament temperature; because tungsten has a positive temperature coefficient of resistance, its resistance increases.”
There is a useful connection to static vs dynamic resistance section:
For a tungsten filament on a V–I graph, the dynamic resistance is the tangent slope, . Since the curve becomes steeper at higher voltage, dynamic resistance increases along the curve.
However, do not automatically say that static resistance V/Iequals the dynamic resistance dV/dI. They are generally different for a curved characteristic.
That makes this bulb example an excellent practical application of the distinction.
Case study 2 — the p-n junction diode
A diode’s V-I graph is dramatically non-linear and, unlike the filament bulb, is not symmetric about the origin:
- Forward bias (V applied in the “easy” direction): almost no current flows until V crosses a threshold (~0.6–0.7 V for silicon), after which current rises very steeply.
- Reverse bias (V applied in the “hard” direction): current remains extremely small (ideally near zero) across a wide range of voltages, until a breakdown voltage is reached.
This isn’t a thermal effect at all — it’s a consequence of the diode’s internal structure (the depletion region at the p-n junction), which behaves fundamentally differently depending on the direction of current flow. A single “resistance” value is almost meaningless for a diode outside a narrow operating range — describing its behavior properly requires the full graph, not a slope.
There is a useful connection to static vs dynamic resistance section:
For a tungsten filament on a V–I graph, the dynamic resistance is the tangent slope, . Since the curve becomes steeper at higher voltage, dynamic resistance increases along the curve.
However, do not automatically say that static resistance V/Iequals the dynamic resistance dV/dI . They are generally different for a curved characteristic.
That makes this bulb example an excellent practical application of the distinction you introduced in the previous section.
Worked Example 1
A component’s V-I graph gives the pairs (V=2V, I=0.5A) and (V=8V, I=1A). Is this component ohmic?
Solution: Compute R at each point:
R₁ = 2/0.5 = 4 Ω
R₂ = 8/1 = 8 Ω
R is not constant across the two points (4Ω vs 8Ω) — the component is non-ohmic. (If it were ohmic, both points would lie on the same straight line through the origin, giving the same R.)
Worked Example 2 — the trap
A student measures R = 4Ω from one point on a filament bulb’s V-I graph (at low voltage) and uses this to predict the current at a much higher voltage, V=20V, getting I = 20/4 = 5A. The actual measured current at V=20V is only 2.5A. What went wrong?
Answer: The student applied a resistance value measured at low voltage (when the filament was relatively cool) to a high-voltage operating point (when the filament has heated up considerably and its resistance has risen well above 4Ω). Since the filament is non-ohmic, R = V/I is not a fixed material constant here — it changes with operating point, and using a single R across a wide voltage range gives an incorrect prediction. The correct approach is to read I directly off the actual graph at V=20V, not to extrapolate using an R measured elsewhere on the curve.
Common errors to watch for
| Error | What’s actually true |
| “A curved V-I graph means the device is broken/faulty” | Non-linearity is completely normal and expected for many real components (bulbs, diodes, transistors) — it’s a feature of the physics, not a defect |
| Extrapolating a single R value across a non-ohmic graph | R = V/I must be evaluated locally, at the specific operating point of interest, for a non-ohmic device |
| Assuming all non-linear graphs have the same cause | The filament bulb’s non-linearity is thermal (R rises with self-heating); the diode’s non-linearity is due to internal junction physics — same symptom (curved graph), different mechanisms |
| Believing a straight line automatically means “ohmic” | Only true if the line passes through the origin. A straight line offset from the origin still implies R = V/I varies with the operating point, even though |
Quick Practice
Q1. A resistor’s V-I graph is a straight line through the origin, and a diode’s V-I graph is a steep curve that’s nearly flat (zero current) for a wide range of reverse voltages. Which one is ohmic, and how do you know from the graph alone?
Answer: The resistor is ohmic — straight line through the origin means constant R = V/I at every point. The diode is clearly non-ohmic since its graph is neither straight nor through the origin in a simple way.
Q2. Why does a filament bulb’s resistance calculated from its rated (operating) voltage and current differ significantly from its resistance measured cold with a simple ohmmeter?
Answer: The ohmmeter measurement is taken with the filament at room temperature (cold), while the rated operating values correspond to the filament glowing at full brightness, much hotter. Since resistance rises with temperature for a metal filament, the “hot” resistance at rated operation is significantly higher than the “cold” resistance.
Q3. Sketch (in words) how you’d expect the V-I graph of two identical ohmic resistors of different fixed values, R₁ < R₂, to compare on the same set of axes.
Answer: Both would be straight lines through the origin, but the line for R₁ (smaller resistance) would be steeper when I is plotted on the y-axis against V on the x-axis (larger current for the same voltage) — or equivalently, less steep if V is plotted against I, since a smaller R means a smaller slope in the V-vs-I convention.
Summary
Ohmic vs Thermal Non-Ohmic vs Junction Non-Ohmic Devices
| Feature | Ohmic (e.g. fixed resistor) |
Non-ohmic, thermal (e.g. tungsten filament bulb) |
Non-ohmic, junction (e.g. p–n diode) |
| V–I graph shape | Straight line through the origin | Curved; becomes progressively steeper as the filament heats | Strongly non-linear and highly asymmetric |
| V–I relationship | V is proportional to I | V is not proportional to I | V is not proportional to I |
| Static resistance, R = V/I | Constant | Changes with operating point; generally, increases as the filament gets hotter | Changes dramatically with operating point and bias direction |
| Dynamic resistance, r = dV/dI | Constant and equal to R | Changes along the curve and generally increases as temperature rises | Changes strongly along the characteristic; can be very small in forward conduction and very large in reverse bias before breakdown |
| Forward-bias behaviour | Same basic relationship in either direction | Similar behaviour in either direction, assuming the same thermal conditions | Current remains small initially, then rises rapidly after the forward-conduction region is reached |
| Reverse-bias behaviour | Follows the same linear relationship with reversed polarity | Approximately similar magnitude behaviour for positive and negative current, subject to thermal effects | Very small reverse current until breakdown is reached |
| Symmetry about the origin | Approximately symmetric | Approximately symmetric if thermal conditions are comparable | Strongly asymmetric |
| Main cause of behaviour | Resistance remains approximately constant | Temperature-dependent resistance caused by self-heating | p–n junction physics, including the depletion region and charge-carrier behaviour |
| Effect of increasing current | Resistance remains approximately unchanged | Heating increases → temperature rises → resistance increases | Junction current changes non-linearly; behaviour depends strongly on bias |
| Temperature dependence | Usually small over the normal operating range | Strong; central to the non-ohmic behaviour | Important; temperature affects forward voltage and reverse leakage, but temperature change is not the fundamental cause of diode non-linearity |
| Breakdown region | Not normally a defining feature | Not normally a defining feature | Reverse current can increase sharply after the breakdown voltage is reached |
| Can one constant R represent the device? | Yes, within its specified operating range | No, over a wide operating range | No; the operating point and bias direction must be specified |
| Typical application/example | Current-limiting resistor, circuit resistor | Incandescent lamp | Rectification, switching, signal detection, voltage regulation using suitable diode types |
The key comparison
Ohmic resistor:
Constant resistance → straight V–I line
Tungsten filament:
Self-heating → temperature rises → resistance rises → curved V–I line
p–n diode:
Junction physics → strongly different forward and reverse behaviour → asymmetric, non-linear V–I characteristic
Remember
For a V–I graph:
Static resistance: R = V/I
Dynamic resistance: r = dV/dI
For an ohmic resistor, these are constant and equal.
For non-ohmic devices, they generally vary with the operating point.
The shape of the V–I graph is therefore not just a picture — it tells you how the device’s resistance and electrical behaviour change as the operating conditions change.
https://edu-physics.com/current-electricity-master-hub/: V-I Characteristics: Ohmic and Non-Ohmic Conductors
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