Parallel and mixed connection

One lamp is switched off while another stays lit. Why does the break no longer stop the whole circuit? Trace each branch and find what they share.

A node is a connection of wires where current can split. In parallel, both ends of each element connect to the same pair of nodes. Several paths exist between that pair. Draw longer wires or rotate a branch on paper: the connection type stays the same if the links are preserved. Read a circuit by tracing where terminals connect, not by how close symbols are on the page.

Voltage describes a pair of points. Parallel branches share their start and end, so U = U₁ = U₂, in volts. This does not require equal resistances: the resistors may be very different. If the source changes the voltage between the nodes, every branch voltage changes equally. Connecting wires are ideal in this model. If their resistance matters, different points on real long wires cannot automatically be treated as one electrical node.

In steady operation charge does not accumulate at the node. Coulombs arriving per second in the common wire equal the total leaving through the branches: I = I₁ + I₂, in amperes. At a common voltage, I₁ = U/R₁ and I₂ = U/R₂. A smaller R carries more current; equal resistances share current equally. If one branch current becomes zero, the total equals the remaining branch current.

Equivalent resistance is one resistor that carries the same total current as the whole group at the same voltage. Add another branch while keeping U unchanged. The old path remains, and the new one adds current, so total I rises. From R = U/I, total resistance falls. This explains the surprising result: adding a resistor can reduce R if it adds a parallel path.

Write the same reasoning exactly: U/R = U/R₁ + U/R₂. For nonzero U, cancel it to obtain 1/R = 1/R₁ + 1/R₂. Reciprocal resistances add because currents add at the same voltage. For finite positive resistances, total R is smaller than each one. As one branch resistance becomes very large, its reciprocal and current approach zero, so the remaining branch almost entirely determines the result.

For two resistors, putting the fractions over a common denominator gives R = R₁R₂/(R₁ + R₂). Test identical branches R₀: the result is R₀/2 because there are now two identical paths. A very low-resistance branch also makes total R small. The product-over-sum formula applies only to two branches; for three or more, return to the sum of reciprocals or combine groups step by step.

In a series-parallel circuit, first identify a simple group with two clear terminals. Replace a parallel pair by one equivalent resistance. If a single resistor is in series before it, add that R. After finding total current, return to the original circuit: all current passes through the series element, then splits. The parallel pair's branches have equal voltage, but it need not equal the whole source voltage.

Open one parallel branch: the other still links both nodes, so a path through the source remains. If the source maintains its voltage and the remaining branch resistance is unchanged, that branch's current stays the same. Total source current falls by the disconnected branch's current. A real source's voltage can depend on load, so complete independence belongs to the fixed-U model, not every battery.

R₁ = 6 Ом and R₂ = 3 Ом are connected in parallel across 12 V. Find the currents and total R.

Both resistors connect to the same node pair, so each has 12 V across it. Calculate separately: I₁ = 12/6 = 2 А, I₂ = 12/3 = 4 А.

The common wire carries the sum: I = 2 + 4 = 6 А. Equivalent resistance must give this 6 A at 12 V, so R = 12/6 = 2 Ом.

The result is smaller than both 3 and 6 ohms. The extra path increases current; the branch with half the resistance carries twice the current.

I₁ = 2 А, I₂ = 4 А, I = 6 А, R = 2 Ом. The common wire carries the combined charge flow of both branches.

Parallel connection is defined by a shared node pair; equal voltage follows from the shared pair of points.

Charge conservation gives the current sum; Ohm's law turns it into a sum of reciprocal resistances.

An extra branch reduces equivalent resistance; independent branch currents require unchanged source voltage.

Knowing voltage and current, we can calculate electrical energy use.