Circuit Analysis (Equation) with Multiple Voltage Sources

[ThLiOp]

Homework Statement

Derive the differential equation that relates the voltage (V) to the current (I) entering the circuit.

Homework Equations

Let:

i_c = C dV/dt = current traveling through the capacitor (C)
i_k = (V+E_k) / R_k = current traveling through the potassium (K) channel
i_Na = (V-E_Na) / R_Na = current traveling through the sodium (Na) channel
i_Cl = (V+E_Cl) / R_Cl = current traveling thought the chlorine (Cl) channel

I = i_c + i_k + i_Na + i_Cl = total current

The Attempt at a Solution

I = i_c + i_k + i_Na + i_Cl
I = C dV/dt + (V+E_k) / R_k + (V-E_Na) / R_Na + (V+E_Cl) / R_Cl

My confusion is from the signs of the batteries. What is the convention (or physical meaning) when you are subtracting the voltages. For example, in the potassium (K) channel, it is V - (-E_k); while in the sodium (Na) channel, the signs on the battery are reversed. How do I know when a voltage source is positive or negative in relation to the current flow?

Thank you!

 

[gleem]

Your equations look correct.

The voltage polarities are usually given. They are usually the consequence of some physical (generator) or chemical (battery) situation. When the current which is conventionally considered a positive charge flow is taken from the negative side of a voltage source to the positive side the voltage drop is taken as negative while if the current is "flowing" from positive to negative the voltage drop is taken as positive

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[CWatters]

My confusion is from the signs of the batteries. What is the convention (or physical meaning) when you are subtracting the voltages. For example, in the potassium (K) channel, it is V - (-Ek); while in the sodium (Na) channel, the signs on the battery are reversed. How do I know when a voltage source is positive or negative in relation to the current flow?

You didn't say how you arrived at your equations for I_k, I_NA and I_CL?

One way is to use KVL. KVL states that the voltages around a loop sum to zero. However before you can apply KVL you have to define which direction you mean by +ve current. Let's do KVL for the potassium channel...

First I arbitrarily define +ve I_K as current flowing down through R_k.
Then I arbitrarily decide to sum the voltages around the loop clockwise and get...

V + (-I_kR_k) + E_K = 0

Rearrange and you get..

I_K = (E_K + V)/R_K

which is the same as you got.

The essential thing is to mark the diagram with arrows showing your definition of +ve current at the outset and stick to it when you write the equations. It doesn't matter which direction you choose as +ve just as long as you are consistent when you write the equations. eg If you assume +ve is down through R_K then there will be a voltage drop in R_K when you travel clockwise summing the voltages. That's why I wrote -I_kR_k. If I had defined +ve current as flowing up through R_K then I would have written +I_kR_k in the equation.

In short it doesn't matter which direction you choose as +ve current flow OR which way around a loop you sum the voltages as long as you are consistent. It all comes out in the wash.

Try it for the other branches.

 

[CWatters]

PS Your equations are correct if you defined +ve current flow as down through the relevant resistor. Suppose you solved all the equations and one came out as -ve. What would that mean if you haven't defined which direction is +ve?