Internal Resistance of a Real Battery + Resistance of the Circuit

In summary: At the other extreme, with no load resistor, there is no heat dissipated. So there is a maximum somewhere in between, and that is what you are being asked to find.In summary, the question asks to show that the rate at which energy is dissipated in R as thermal energy is a maximum when R = r, and that this maximum power is P = EMF2/4r. By applying the maximum power theorem, it can be seen that the maximum heat dissipated occurs at r = R, rather than at an ideal battery with zero internal resistance.
  • #1
ohgeecsea
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Homework Statement


(a) In the figure shown, show that the rate at which energy is dissipated in R as thermal energy is a maximum when R = r. (b) Show that this maximum power is P = EMF2/4r.

http://www.practicalphysics.org/imageLibrary/jpeg400/208.jpg [Broken]

The only difference between this picture and the one in my book is that they specify the direction of current going one way (clockwise) throughout the whole circuit.



Homework Equations



P = i2r

EMF - ir - iR = 0

P = EMF2/4r



The Attempt at a Solution



(b) switching EMF - ir - iR = 0 so that it is equal to i:

i = EMF / (r + R)

and assuming r = R:

i = EMF / 2r

substituting this in for i in P = i2r, you get:

P = [ EMF / 2r ]2r
P = EMF2r/4r2
P = EMF2/4r

(a) for r = R, as shown above:

P = EMF2/4r

and for r =/= R:

P = EMF2r/(r2 + 2rR + R2)

Thing is, I don't really understand how the rate of dissipation of thermal energy is greatest at r = R. Isn't this just a linear relationship? Wouldn't the power be greatest if the internal resistance was zero, as in an ideal battery?
 
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  • #2
See "maximum power theorem" at wikipedia.

Wouldn't the power be greatest if the internal resistance was zero, as in an ideal battery?

Max power would be delivered to the load resistor if internal resistance were zero, but the question is about maximum heat dissipated.
 
  • #3


Your calculations are correct, but your interpretation of the results needs some clarification.

Firstly, it is important to note that the maximum power dissipation occurs when the internal resistance of the battery is equal to the resistance of the circuit, not necessarily when r = R. This can be seen from the equation P = EMF2/4r, where the power is maximized when r = R.

To understand why this is the case, let's consider the two extreme scenarios - when r = 0 (ideal battery) and when r = ∞ (open circuit).

In the case of an ideal battery, the internal resistance is zero, meaning all the energy is transferred to the external circuit and there is no dissipation of thermal energy. This may seem like the maximum power, but it is important to remember that in a real battery, some energy is always lost due to the internal resistance.

On the other hand, in the case of an open circuit, there is no current flowing and therefore no power dissipation. So, the power is also zero in this case.

Now, when r = R, the internal resistance and external resistance are equal, meaning the energy transfer is balanced between the battery and the external circuit. This results in the maximum power dissipation, as any deviation from this balance would result in either more energy being lost in the internal resistance (when r < R) or less energy being transferred to the external circuit (when r > R).

In conclusion, the maximum power dissipation occurs when the internal resistance of the battery is equal to the resistance of the circuit because it is the point of balance between energy transfer and energy loss.
 

1. What is internal resistance of a battery?

The internal resistance of a battery refers to the resistance that exists within the battery itself. It is caused by the chemical reactions that occur within the battery and can vary depending on factors such as the type of battery and its age.

2. How does internal resistance affect a battery's performance?

Internal resistance can affect a battery's performance by causing a voltage drop within the battery. This means that the battery's output voltage will be lower than its open circuit voltage, resulting in decreased performance and a shorter lifespan.

3. What is the role of resistance in a circuit with a battery?

Resistance in a circuit with a battery is important as it limits the flow of current through the circuit. This is necessary to prevent damage to the components in the circuit and to control the amount of power being delivered by the battery.

4. How does the resistance of the circuit affect the overall performance of a battery?

The resistance of the circuit can affect the overall performance of a battery by determining the amount of current that can flow through the circuit. A higher resistance means less current can flow, resulting in decreased performance and a slower discharge of the battery.

5. Can internal resistance be reduced in a battery?

While it is not possible to completely eliminate internal resistance in a battery, it can be reduced by using materials with lower resistance, improving the design of the battery, and avoiding extreme temperatures which can increase internal resistance.

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