Power in a simple electric circuit

[Karol]

Homework Statement

A circuit consists of a battery with internal resistance r and a resistor with resistance R.
It has been found that the maximum power generated is when R=r.
Prove that in those conditions $P_{max}=\frac{ε^2}{4r}$

Homework Equations

Power invested in a resistor: P=i²R
Power invested between 2 points in a circuit: P=Vi

The Attempt at a Solution

The current: $i=\frac{V}{2r}$
Voltage drop on the internal resistance: $V=ir=\frac{V}{2}$
Power between the terminals of the battery: $P=iV=\frac{V}{2r} \frac{V}{2}=\frac{V^2}{4r}$
But the power loss on one resistor (the internal or the external):
$$P=i^2r=\frac{V^2}{4r^2}\cdot r=\frac{V^2}{4r}$$
It's identical to the total power, and there are 2 resistors so they consume double the power, how come?

 

[Doc Al]

It's identical to the total power, and there are 2 resistors so they consume double the power, how come?

The total power delivered by the battery is the current times the battery EMF (which is V), not the terminal voltage (V/2).

 

[Karol]

So:
$$P_{max}=\varepsilon\cdot i=\varepsilon\cdot \frac{\varepsilon}{2r}=\frac{\varepsilon^2}{2r}$$
The book asked about the maximum power of the battery and i guess this is the answer. the answer of the book is:
$$P_{max}=\frac{\varepsilon^2}{4r}$$

 

[Dick]

So:
$$P_{max}=\varepsilon\cdot i=\varepsilon\cdot \frac{\varepsilon}{2r}=\frac{\varepsilon^2}{2r}$$
The book asked about the maximum power of the battery and i guess this is the answer. the answer of the book is:
$$P_{max}=\frac{\varepsilon^2}{4r}$$

The maximum power the question is asking about is the maximum power delivered to the load, that's the external resistor only.

 

[Karol]

Thanks, i guess you are right. but is there, in general, a meaning to the multiplication of the terminals of the battery voltage and the current P=V⋅i? not in this specific problem but in general. i guess it's the power delivered to the rest of the circuit, right?

 

[NascentOxygen]

Thanks, i guess you are right. but is there, in general, a meaning to the multiplication of the terminals of the battery voltage and the current P=V⋅i? not in this specific problem but in general. i guess it's the power delivered to the rest of the circuit, right?

Yes. That's the power that the battery is delivering [to the external circuit]. In this problem, and in general.

 

[Karol]

Je Suis Charlie also, thanks