Suppose two wires in a network join the same pair of nodes. Explain why their effect on the rest

Suppose two wires in a network join the same pair of nodes....

Key Concepts

Resistance

Resistance represents the opposition to the flow of electrical current. When resistors are connected in parallel, their combined effect is given by the reciprocal of the sum of their individual reciprocals, meaning that the overall resistance is less than any individual resistance. This inverse relationship with conductance facilitates analysis of complex electrical networks.

Parallel Circuit Analysis

In a parallel configuration, electrical components share the same two nodes, which means they are exposed to the same voltage. This fact leads to the characteristic that the total conductance is the sum of the individual conductances, as each parallel branch independently contributes to the overall current flow.

Conductance

Conductance is a measure of how easily current flows through an electrical component, and it is defined as the reciprocal of resistance. In systems where components are connected in parallel, their ability to conduct current adds up, allowing for a straightforward calculation of the total conductance by summing the individual values.

Transcript

00:01 So this is a question of resistivity, and we'll begin by writing down what we do know.

00:05 The combination resistance is equal to 10 kilo -ooms, and it's equal to the sum of the individual resistances or the individual wires because they are in series.

00:15 So we get the first equation.

00:17 Next, we'll be looking at the equation that we'll use to solve the question.

00:20 So are the resistance at a given temperature is equal to the reference temperature times 1 plus alpha, which is the temperature's coefficient of, resistivity, which changes based on the substance of the wires times delta t, which is the difference in the reference temperature and the temperature you were solving for.

00:40 So since there is a variable based on the substance, we'll write down two equations.

00:45 So the first wires resistance will become r10 times 1 plus alpha 1, which is its coefficient of resistivity, times delta t.

00:58 And the second wire, the nicrome, will do similarly.

01:03 R21 plus alpha 2 because there are different temperature coefficients of resistivity times delta t.

01:09 Now we know that the equivalent or the combination resistance is equal to the sum of the individuals.

01:15 So we get the combination resistance, the final one with the change temperature, is equal to the sum of the initial two equations.

01:22 So i'm just using the right hand side of the equations and writing them as a sum to get the combination resistance.

01:32 And make sure you have two different alphas because they are not the same.

01:36 Next, i'll just be expanding the terms.

01:39 So we get r10 plus r10 plus this part is factored.

01:44 So we get r10 alpha 1 plus r2 alpha 2, simply because the delta t's are the same and it can be taken out like that.

01:54 So next we come under the question of, how do we solve the rest of this question? we have a bunch of variables.

02:00 So we know that the r equivalent final must also equal 10 kilo -ooms because we want the combination resistance to say the same, regardless of a change in temperature.

02:11 So delta t is a variable that will be changing, and we don't know how much it changes.

02:16 And r10 plus r20 is already equal to 10 kilooms as seen by our very first given fact...

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