Just Can't Get Enough Nodal Analysis

The questions below are due on Monday September 16, 2024; 10:00:00 PM.
 
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Solve for the given quantities below by using the node method (and/or series/parallel simplifications where possible).

Don't let the triangly-ness of the circuits mess you up; what matters is not the angle at which each component is drawn, but simply what nodes it's connected to (and what other components are connected there).

For this equation, enter all of your answers as exact decimal numbers or as fractions. Simple expressions are allowed using standard operations (+, -, /, *).

1) Circuit 1

Here is a circuit, with a reference node and two unknown node potentials labeled, and where all resistors have a resistance of 1{\rm k}\Omega:

What is the potential at node e_1, relative to the indicated reference, in Volts?

What is the potential at node e_2, relative to the indicated reference, in Volts?

Note that the voltage drop across the current source is given by e_1 - e_2. What is this voltage drop, in Volts?

2) Circuit 2

Now let's consider a little twist on this circuit, adding a few more components (and now assigning different labels to the node potentials). As with the circuit above, all resistors are 1{\rm k}\Omega.

What is the potential at node e_1, relative to the indicated reference, in Volts?

What is the potential at node e_2, relative to the indicated reference, in Volts?

What is the potential at node e_3, relative to the indicated reference, in Volts?

Note that the voltage drop across the current source is given by e_1 - e_3. What is this voltage drop, in Volts?