"Electric Current"

About

When charges move, we have electric current — the lifeblood of electrical circuits. This chapter introduces the fundamentals of current electricity: Ohm's law, resistance and resistivity, the colour coding of resistors, electromotive force (EMF), and powerful circuit analysis tools like Kirchhoff's laws and the Wheatstone bridge.

Key Concepts

17.1 Electric Current

Electric current is the rate of flow of charge:

$I=\frac{Q}{t}$

SI unit: ampere (A) = C/s.

Conventional current flows from positive to negative — opposite to the direction of electron flow.

If $n$ electrons pass per second: $I=ne$, where $e=1.6\times 10^{-19}$ C.

17.2 Ohm's Law

$V=IR$

For ohmic conductors (metals at constant temperature), $V\propto I$ — the V-I graph is a straight line through the origin.

Ohmic conductors: Pure metals (Cu, Ag, Al), metallic alloys (constantan, manganin).

Non-ohmic conductors: Diodes, transistors, filament bulbs (V-I is non-linear because resistance changes with temperature/voltage).

17.3 Resistance and Resistivity

Resistance: $R=\frac{V}{I}$. SI unit: ohm (Ω).

$R=\rho \frac{l}{A}$

Resistivity ($\rho$) is a material property:

• Depends on: nature of material, temperature
• Does NOT depend on: length, area of cross-section, shape
• SI unit: Ω⋅m

Conductivity: $\sigma =1/\rho$

Effect of changing dimensions (same material, same voltage):

• Length doubled ($l\to 2l$) → $R$ doubles → $I$ halves
• Area halved ($A\to A/2$) → $R$ doubles → $I$ halves

17.4 Colour Code for Resistors

Gold (tolerance ±5%), Silver (±10%), No band (±20%).

17.5 Combinations of Resistors

Series: $R_{\text{eq}}=R_1+R_2+R_3+\dots$

• Same current through each
• Voltage divides

Parallel: $\frac{1}{R_{\text{eq}}}=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}+\dots$

• Same voltage across each
• Current divides

17.6 EMF and Internal Resistance

EMF ($\mathcal{E}$) is the energy supplied per unit charge by a source.

$V=\mathcal{E}-Ir$

Where $r$ = internal resistance, $I$ = current drawn.

17.7 Kirchhoff's Laws

First Law (Current Law / KCL): At any junction, $\sum I_{\text{in}}=\sum I_{\text{out}}$. Based on conservation of charge.

Second Law (Voltage Law / KVL): Around any closed loop, $\sum \mathcal{E}=\sum IR$. Based on conservation of energy.

17.8 Wheatstone Bridge

A balanced Wheatstone bridge:

$\frac{R_1}{R_2}=\frac{R_3}{R_4}$

When balanced, the galvanometer current is zero — used to measure unknown resistance.

INTEXT QUESTIONS 17.1

Q1. (a) A current I is established in a copper wire of length l. If the length of the wire is doubled, calculate the current due to the same cell.

Ans: Same cell → same $V$. $R=\rho l/A$. Length doubled → $R_2=2R_1$.

$I_2=V/R_2=V/(2R_1)=I_1/2$. Current reduces to half.

(b) What happens to current if the area of cross-section is decreased to half?

Ans: $A_2=A/2$ → $R_2=\rho l/(A/2)=2R_1$. $I_2=V/(2R_1)=I_1/2$. Current reduces to half.

Q2. The resistivity of a wire of length l and area A is 2 × 10⁻⁸ Ωm. What will be the resistivity of the same metallic wire of length 2l and area 2A?

Ans: Resistivity is a material property — it does NOT depend on length or area. ρ = 2 × 10⁻⁸ Ωm (unchanged).

Q3. A potential difference of 8 V is applied across a wire of length 3 m and area 2 cm². The current is 0.15 A. Calculate resistance and resistivity.

Ans: $R=V/I=8/0.15=53.33$ Ω

$\rho =RA/l=(53.33\times 2\times 10^{-4})/3=3.55\times 10^{-3}$ Ω⋅m

Q4. Do all conductors obey Ohm's law? Give examples.

Ans: No. Ohmic: Pure metals (Cu, Ag), alloys (constantan, manganin). Non-ohmic: Diodes, transistors, filament bulbs (resistance changes with temperature/voltage, V-I is non-linear).

Q5. 5 × 10¹⁷ electrons pass through a cross-section per second from left to right. Determine the value and direction of current.

Ans: $I=ne=5\times 10^{17}\times 1.6\times 10^{-19}=8.0\times 10^{-2}$ A = 0.08 A

Conventional current direction is opposite to electron flow: right to left.

Terminal Exercise

1. Define electric current. Distinguish between conventional current and electron flow.

2. State Ohm's law. Draw the V-I characteristics of an ohmic and a non-ohmic conductor.

3. Define resistance and resistivity. Derive $R=\rho l/A$. On what factors does resistivity depend?

4. Derive expressions for equivalent resistance when resistors are connected in (a) series, (b) parallel.

5. Explain the colour coding of carbon resistors. What is the resistance of a resistor with bands: yellow, violet, orange, gold?

6. State Kirchhoff's laws. Apply them to derive the condition for a balanced Wheatstone bridge.

7. A wire of resistance 10 Ω is stretched to twice its original length. Find the new resistance.

8. Three resistors of 2 Ω, 3 Ω, and 6 Ω are connected in parallel. Find the equivalent resistance. If connected to a 6 V battery, find the current through each resistor.

9. Explain the difference between EMF and terminal potential difference. Derive $V=\mathcal{E}-Ir$.

10. A cell of EMF 2 V and internal resistance 0.1 Ω is connected to a 3.9 Ω external resistor. Find the current and terminal voltage.

11. Using Kirchhoff's laws, find the currents in each branch of a simple two-loop circuit.

12. A potential difference of 10 V is applied across a conductor of length 2 m and area 0.5 mm². If the current is 2 A, find the resistivity of the material.

Worked Examples

Example 1: Ohm's Law

Problem: A 12 V battery is connected across a 6 Ω resistor. Find the current.

Solution: $I=V/R=12/6=2$ A

Example 2: Resistance from Dimensions

Problem: A copper wire ($\rho =1.7\times 10^{-8}$ Ω⋅m) is 10 m long and has radius 1 mm. Find its resistance.

Solution: $A=\pi r^2=3.14\times 10^{-6}$ m²

$R=\rho l/A=1.7\times 10^{-8}\times 10/(3.14\times 10^{-6})=0.054$ Ω

Example 3: Parallel Resistors

Problem: Find the equivalent resistance of 4 Ω and 12 Ω in parallel.

Solution: $1/R=1/4+1/12=3/12+1/12=4/12=1/3$ → $R=3$ Ω

Common Mistakes

1. Confusing resistivity with resistance: Resistivity is a material property; resistance also depends on geometry.
2. Applying Ohm's law to non-ohmic devices: $V=IR$ always defines $R$, but $R$ may not be constant.
3. Forgetting that stretched wire changes both length AND area: When length doubles, area halves → $R$ becomes 4×.
4. Mixing up series and parallel formulas: Series → resistances add. Parallel → reciprocals add.
5. Ignoring the internal resistance of a cell: Terminal voltage = EMF − $Ir$.

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