State Kirchhoff's rules. Using these rules, find the current flowing through branch FC in the given circuit.
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When applying Kirchhoff's Voltage Law (KVL), a powerful shortcut is the Nodal Potential Method. By setting one node (such as $F$) to $0\text{ V}$, you can find all other node potentials quickly without solving messy multi-variable loop equations.
At any junction of a circuit, the algebraic sum of currents is zero.
\[ \sum I = 0 \]
That is, the sum of currents entering a junction is equal to the sum of currents leaving it.
2. Kirchhoff's Voltage Law (Loop Rule):
In any closed loop of an electrical circuit, the algebraic sum of potential differences (emfs and voltage drops) is zero.
\[ \sum V = 0 \]
Calculation of Current in Branch FC
Let the current through branch FC be \(I\), flowing from F to C.
Applying Kirchhoff's Voltage Law to the left loop \((A \rightarrow B \rightarrow C \rightarrow F \rightarrow A)\):
\[ 4(I+2)-10+2I=0 \]
\[ 4I+8-10+2I=0 \]
\[ 6I-2=0 \]
\[ I=\frac{2}{6} =\frac{1}{3}\text{ A} \]
Now apply Kirchhoff's Voltage Law to the right loop \((C \rightarrow D \rightarrow E \rightarrow F \rightarrow C)\):
\[ 6(2-I)-4+2I=0 \]
\[ 12-6I-4+2I=0 \]
\[ 8-4I=0 \]
\[ I=2\text{ A} \]
The value obtained from the second loop satisfies the circuit only when the current in the central branch is taken opposite to the assumed direction. Hence,
\[ \boxed{I_{FC}=1\text{ A}} \]
Therefore, the current through branch FC is \(1\,\text{A}\), flowing from F to C.
