# What is Kirchhoff's Current Law

1. Jul 24, 2014

### Greg Bernhardt

Definition/Summary

KCL (KIRCHHOFF'S CURRENT LAW), for constant charge density:

Current in = current out (at a point):

For constant charge density (ie, without capacitors), the total electric current entering a point equals the total electric current leaving it.

KCL (KIRCHHOFF'S CURRENT LAW), at a capacitor:

Current in = displacement current out ($\partial\mathbf{D}/\partial t$).

This must be supplemented by CONSERVATION OF CHARGE for capacitors in series:

For two capacitors in series (that is, where the charge between them is "stuck", with nowhere else to go), the charge on the "facing" sides of the two capacitors must be equal and opposite, and so the charge across both capacitors must be equal ($Q_1\,=\,Q_2$).

KVL (KIRCHHOFF'S VOLTAGE LAW), without inductors:

Potential difference is additive, and around a loop is zero:

The electric potential difference between two points on any electric path is the sum of the potential differences across all the sections of that path ($\sum V\,=\,V_{total}$).

In particular, if the path is a loop, the sum of the potential differences across all the sections of the loop is zero ($\sum V\,=\,0$).

"Potential difference" includes the potential gain (or "voltage" or "emf") from a battery and the potential drop (or "voltage drop") across a resistor or capacitor.

KVL (KIRCHHOFF'S VOLTAGE LAW), with inductors:

Potential difference cannot be defined (because electric field around a loop is not then conservative) when an inductor is present and the current is changing.

However, KVL still works if minus the emf of the inductor ($-\mathcal{E} = LdI/dt$) is deemed to be the potential difference across it: of course, this depends not on the current but only on the rate of change of the current.

On sudden ("implusive") insertion of an inductor into a loop, all components except inductors may be ignored: with no other inductor present, the current through it will not instantaneously change ($\Delta I = 0$): with others, the instantaneous changes ($\Delta I_i$) will be such that the sum of the "impulsive emfs" must be zero ($\Sigma L_i\Delta I_i\ =\ 0$).

Equations

POTENTIAL DROPS:

Across a resistance $R$:

$$V = IR$$, where $I$ is the current (Ohm's Law).

Across a length of wire:

usually so small that it can be ignored.

Across a capacitor $C$:

$V = Q/C$, where $Q$ is the charge.

Across two capacitors $C_1$ and $C_2$ in series:

$$Q = V_1C_1 = V_2C_2 = \frac{(V_2C_2)C_1 + (V_1C_1)C_2}{C_1 + C_2} = (V_1\,+\,V_2)\frac{C_1C_2}{C_1 + C_2} = (V_1\,+\,V_2)\left(\frac{1}{\frac{1}{C_1}\,+\,\frac{1}{C_2}}\right)$$

$$V = V_1\,+\,V_2 = Q\left(\frac{1}{\frac{1}{C_1}\,+\,\frac{1}{C_2}}\right)$$

EMF:

Across an inductor $L$:

$$-\mathcal{E}\ =\ L\frac{dI}{dt}$$

Extended explanation

Alternative names:

KCL/Kirchhoff's current law/Kirchhoff's junction rule/Kirchhoff's first rule.
KVL/Kirchhoff's voltage law/Kirchhoff's loop rule/Kirchhoff's second rule.

Labelling the currents:

In a complicated diagram, there will be different currents through different sections. It is important to label each section with a name for the current such as I1 (or the actual amount if known), and an arrow showing the direction of the current.

It does not matter if any arrow is the wrong way round: this will simply result in a negative value for the current when the equations are solved.

Junctions:

At most points in a circuit, there is no junction, and then Kirchoff's first rule simply says that the current is the same on either side of that point.

Electric path:

An electric path or circuit may have breaks in it if an accumulation of charge on one side of the break can affect charge on the other side.

Such a break is a capacitor.

Capacitors:

When a steady potential difference (voltage) is first applied across a capacitor, the charge density will change, with equal positive and negative charge building up on its two plates until a maximum is reached.

The capacitance, $C$, of the capacitor is the ratio of that maximum charge to the applied voltage: $C\,=\, Q/V$.

Displacement current:

In a dielectric, the charges in the material of the dielectric may be displaced when a current is applied.

The displacement current, $\partial\bold{D}/\partial t$, equals the drop in the current across the dielectric: this inclusion of displacement current preserves the validity of KCL at a capacitor.

An independent set of loops:

An electric system usually contains many loops, and in applying KVL, one chooses the most convenient independent set.

For example, the letter "B" contains three loops (not just two), and KVL applies to each of the three loops, but only two of the equations for those loops are independent.

A switch:

A switch or other gap (that is, where charge on one side does not affect charge on the other side) may be considered as a capacitor with zero $C$ and zero $Q$, so that the potential difference $V\,=\,Q/C\,=\,0/0$ may be anything.

Inductors:

When the current is constant, an inductor behaves like an ordinary wire.

When the current changes, an inductor creates (induces) an opposing emf ($\mathcal{E} = -LdI/dt$).

This may be added to the emf (if any) on the RHS of KVL, but it is more usual to add minus the opposing emf ($-\mathcal{E} = LdI/dt$) to the voltage drops on the LHS.

"Impulsive" insertion of an inductor:

When an inductor is inserted into a circuit with no other inductors present, the current (if any) through the inductor cannot instantaneously change (though it would through any other component), since there is a force (emf) opposing this ($\Delta I = 0$).

For example, in a loop with a battery V and a resistance R, KVL before will give RI = V: if the current almost instantaneously changes from I to I + ∆I (through the whole loop including the inductor) in a very short time δt, KVL would give R(I + ∆I) + L∆I/δt = V, or R∆I + L∆I/δt = 0, ie ∆I = Rδt/L or approximately zero.

Of course, a component cannot strictly speaking be just "popped in" to a loop: in practice, it must be placed in series with a huge resistance but in parallel to a section of wire of almost zero resistance: then the sizes of the resistances are suddenly changed so that virtually the whole current goes through the inductor instead of the original section: KVL for the original loop can then be compared with KVL for the new loop via the inductor.

With another inductor present, the current through both inductors must be the same, so it must instantaneously change: the changes will be opposite, and the two forces opposing this will be equal and opposite: in other words, the "impulsive changes in current" ($\Delta I_i$) will be such that the sum of the "impulsive emfs" must be zero ($\Sigma L_i\Delta I_i\ =\ 0$).

For example, in a loop with a battery V a resistance R an inductor L and a steady current I, and inserting a second inductor L2 through which a current I2 was already flowing, KVL before will again give RI = V: if the current almost instantaneously changes to J (through the whole loop including both inductors) in a very short time δt, KVL would give RJ + L(J - I)/δt + L2(J - I2)/δt = V, or R(J - I) + L(J - I)/δt + L2(J - I2)/δt = 0, or approximately L(J - I) + L2(J - I2) = 0.

Components in series:

Round a loop, of course, everything is in series!

For resistors or inductors in series with the same current through them, simply add the resistances or the inductances.

For capacitors in series, however, there is no actual current, and the displacement currents are generally not the same: instead it is the charges that are the same, and this results in the reciprocal law for capacitances (see Equations above).

Kirchhoff's other laws:

There are also Kirchhoff's three laws of spectrosocopy, and Kirchoff's law of thermal radiation.

Kirchhoff, together with Bunsen, discovered caesium (Cs) and rubidium (Rb).

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