MHT-CET Physics · Current Electricity

Kirchhoff's Current and Voltage Laws

At any junction the current flowing in equals the current flowing out (Kirchhoff's first law, conservation of charge), and round any closed loop the e.m.f.s add up to the voltage drops (the second law, conservation of energy); between them they solve any network of cells and resistors.

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Why this matters

19 PYQs, 3 HARD, nearly all read from a circuit diagram. Eight are the junction law — the missing current at a point where several wires meet, and which conservation law each rule rests on; eleven are the loop law — the current in a loop with opposing cells, the potential difference between two points along a branch, two cells sharing a resistor, and the value of R that stops the current in a galvanometer. Two cards.

Concept 1 of 2: The Junction Law

Charge does not pile up at a junction, so whatever flows in must flow out: ΣI_in = ΣI_out. With several wires meeting at a point, add the arrows pointing in, add those pointing out, and the difference is the missing current — in the direction that balances them. In a network, walk from junction to junction carrying the running total. The first law is conservation of charge; the second, the loop law, is conservation of energy.

Definition

$\sum I_{\text{in}} = \sum I_{\text{out}}$ at every junction.
In 10 + 2.5 + 5 A, out 6 A ⇒ the fifth wire carries 11.5 A outward.
Walk a network node by node: 20 A in, 15 A down one side ⇒ 5 A along; add 3 A ⇒ 8 A; ... the last node collects 18 A.
First law ⇒ conservation of charge; second law ⇒ conservation of energy.

Junction law

\sum I_{\text{in}} = \sum I_{\text{out}}

Worked example

At a junction, 3 A and 7 A flow in; 4 A and 1.5 A flow out along two wires. The fifth wire's current?

Practice this conceptself-check · 2 quick reps

The same idea in a real exam question:

MHT-CET · 2024 · 9th May Shift 2 · Q20Moderate

Example 1 · Current Electricity · Kirchhoff's Laws and Circuit Analysis

Five current carrying conductors meet at a point O as shown in figure. The magnitude and direction of the current in conductor OP is

Swapping the conservation laws

The JUNCTION law conserves charge; the LOOP law conserves energy. 'Kirchhoff's second law' is the loop law.

Drill 7 more on the junction law

Concept 2 of 2: The Loop Law and Potential Differences

Walk round a loop and add the changes: through a resistor in the direction of the current the potential falls by IR; through a cell from − to + it rises by E. The total is zero. For the potential difference between two points on a branch, walk from A to B: V_A − V_B = sum of the drops on the way (IR for each resistor, E for each cell entered at its + end). Two cells opposing each other in one loop drive (E₁ − E₂)/R_total. When a branch with its own cell carries no current (a balanced galvanometer), that cell's e.m.f. equals the drop across the resistor it sits across.

Definition

Round a loop: $\sum E = \sum IR$ .
Opposing cells: $I = \dfrac{E_1 - E_2}{R_{\text{total}}}$ (200 V against 10 V with 38 Ω ⇒ 5 A; 100 V against 5 V with 19 Ω ⇒ 5 A).
Along a branch: $V_A - V_B = \sum IR + \sum E$ for cells entered at +: 2 A through 2 Ω, a 3 V cell, 1 Ω ⇒ 9 V; 3 A through 4 Ω, 5 V, 2 Ω ⇒ 23 V.
Galvanometer null: $\dfrac{10R}{4 + R} = 6$ ⇒ R = 6 Ω.
Equal parallel sources feeding a load: combine to one source (3 V, 3 Ω twice ⇒ 3 V, 1.5 Ω; through 6 Ω ⇒ 0.4 A).

Loop law

\sum E = \sum IR

Worked example

A 12 V cell and a 4 V cell oppose each other in a loop with 1 Ω, 1 Ω internal resistances and a 6 Ω resistor. Current, and the voltage across the 6 Ω?

Practice this conceptself-check · 2 quick reps

The same idea in a real exam question:

MHT-CET · 2025 · 22 April Shift II · Q20Moderate

Example 2 · Current Electricity · Kirchhoff's Laws and Circuit Analysis

In the given circuit, current flowing through the circuit i

Getting the sign of a cell wrong along a branch

Going from A to B, a cell entered at its + terminal is a DROP of E; entered at − it is a rise. Draw the path and mark each sign before adding.

Drill 10 more on the loop law and potential differences

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

The Junction Law
Junction law
$\sum I_{\text{in}} = \sum I_{\text{out}}$
The Loop Law and Potential Differences
Loop law
$\sum E = \sum IR$

Watch out for (2)

Swapping the conservation laws
→ The Junction Law
Getting the sign of a cell wrong along a branch
→ The Loop Law and Potential Differences

Test yourself on Current Electricity

20 past MHT-CET questions from this chapter, timed at 18 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Current Electricity chapter test Drill all 19 questions