Energy dissipation across two resistors in series example

A student builds a circuit with a battery and two resistors in series. The resistance of R2 is double the resistance of R1. Below is the graph of the energy lost at R1 over time. So, that's this graph. Which of the following shows the energy lost at R2 over time in terms of E1? We have these four choices here, so pause this video and see if you can figure it out on your own.

All right, now let's do this together. So first, let's visualize the circuit. You have your voltage source; it's our battery here, so it's a positive terminal, negative terminal, and then this could be our first resistor. Then, that's our second resistor. They tell us that these two resistors are in series, and we complete the circuit here. This is going to be R1; this is R2.

If you just have one circuit like this, where you don't have any parts of it that are in parallel, your current throughout this circuit is going to be constant. So, we could say at any point here, our current, let's just call that I. Now they tell us how these resistances relate to each other. If R1 has a resistance of, let's call it X ohms, they tell us that the resistance of R2 is double the resistance of R1. So, this is X ohms; this is going to be 2X ohms.

Now, to think about what would be if this is the energy dissipated across R1 over time, to think about the energy dissipated across R2 over time, let's think about power and energy and what we know about them. So we know that power is equal to energy dissipated over time. We could write, for example, power is equal to energy dissipated over time, which can be expressed as your change in voltage across a resistor times the current going through that resistor.

If you want an expression for energy dissipated over time, you can look at this part right over there. We can just multiply both sides by delta t. So we could write that energy dissipated over time is going to be equal to our change in voltage across the resistor times the current going through the resistor times our change in time.

Now, if we are starting at time equals zero, our change in time is just going to be whatever time we are at minus zero. So, it's going to be whatever time we're at. So I'm just going to write times T right over here. Now your change in voltage across a resistor – we have learned before – your change in voltage across a resistor is just going to be the current going through that resistor times the resistance.

So we could take this and substitute it back for delta V. And what do we get? We get energy dissipated is going to be equal to our current going through a resistor times the resistance times the current going through the resistor times time. Or we could write that energy dissipated as a function of, well, as a function of all three of these things, is going to be the current squared times the resistance times time.

Now, we already know from this that, at time T1, we have an energy dissipated of E1. And that's with the resistance of X ohms. So we can say we have energy dissipated of E1 is equal to I squared times X ohms; that's the resistance times T1.

But now, let's think about what the energy dissipated at time T1 for our second resistor would be. The energy dissipated, if we just go back to this equation right over here, is going to be equal to T1 times our current. Remember, we have the same current going through both resistors; they're in series, so we have the same current there, I squared.

And then, what is going to be the resistance? Well, instead of an X, we have a 2X here. We can rewrite this as 2 times I squared times X times T1. Well, this right over here is the same thing as the amount of energy dissipated at time T1 in our first resistor, so this is E1.

So at time T1, the energy dissipated in our second resistor is going to be twice as much energy. What we want to look at is, at time T1, we want to have twice as much energy. This option only has a fourth as much, so we rule that out. This has half as much energy dissipated; rule that out. This has the same amount of energy dissipated; we rule that out.

But then here, at time T1, we have dissipated twice as much energy, so we like that choice right over there. Another way to think about it is if you have the same current going through two resistors, and the second resistor has twice the resistance, in any given amount of time for the second resistor, you're going to dissipate twice as much energy.

Another thing about it: you'll have twice the slope of this line. So this one right over here clearly has twice the slope of this, or you could say, for any given amount of time, so from time equals 0 to T1, you would dissipate twice as much energy. So instead of E1, you would dissipate 2 times E1.