The idea is to hook nodes A and C up to a battery with a given voltage, say 130 Volts. Then electric current will flow through the various resistors. Intuitively it helps to think of current as a flow from A to C, even though this may not be completely accurate from a physical point of view. The most important relationship between voltage and current is called Ohm’s Law. Namely, suppose that in the circuit we find a resistor (i.e., a lamp, a fridge, etc.) hooked up to nodes P and Q, then:
V(Q) - V(P) = R(PQ) * I(PQ)
In words, the voltage difference V(Q) - V(P) determines the current I(PQ) from P to Q, up to a constant multiplicative factor R(PQ), known as the resistance. Note that the resistance is always a positive number, while the current I(PQ) could be positive or negative. The physical meaning of a negative current from P to Q is simply that the flow is actually going from Q to P.
The other important property of flows is the Node Law. Basically, at every node not hooked to the battery (nodes B and D in this case), the amount flowing in should equal the amount flowing out. By combining Ohm’s Law with the Node Law, one can see that, at all the nodes that are not hooked up to the battery, the voltage V is harmonic, meaning that the value at that node is equal to the average over the values at the neighboring nodes. For instance, the value V(D) is equal to a (weighted) average of the values of V at the neighbors of D. The weights needed to take the average are the edge conductances, which are just the reciprocal of the resistances. For example:
V(D) = V(A) + V(B) + V(C) / 2
Without loss of generality, we will assume that V(A) = 0 and V(C) = 130, since the only thing that matters is the voltage drop. So our unknowns are x1 = V(B) and x2 = V(D).
(a) Equation (1) gives a linear relationship between the variables x1 and x2 by requiring harmonicity at D. Your task is to do the same at node B, then write down a 2-by-2 system of equations and solve it.
(b) Now that you have the voltage, use Ohm’s Law to deduce all the currents.
(c) Finally draw a picture of the network and label all the variables you have found above, including arrows showing positive flows.