A. Load resistor R is attached to a battery of emf ℰ and...

a. Load resistor $R$ is attached to a battery of emf $\mathcal{E}$ and internal resistance $r .$ For what value of the resistance $R,$ in terms of $\mathcal{E}$ and $r$, will the power dissipated by the load resistor be a maximum? b. What is the maximum power that the load can dissipate if the battery has $\mathcal{E}=9.0 \mathrm{V}$ and $r=1.0 \Omega ?$ c. Why should the power dissipated by the load have a maximum value? Explain. Hint: What happens to the power dissipation when $R$ is either very small or very large?

a. Load resistor $R$ is attached to a battery of emf $\mathcal{E}$ and internal resistance $r .$ For what value of the resistance $R,$ in terms of $\mathcal{E}$ and $r$, will the power dissipated by the load resistor be a maximum?
b. What is the maximum power that the load can dissipate if the battery has $\mathcal{E}=9.0 \mathrm{V}$ and $r=1.0 \Omega ?$
c. Why should the power dissipated by the load have a maximum value? Explain.
Hint: What happens to the power dissipation when $R$ is either very small or very large?

Key Concepts

Maximum Power Transfer Theorem

The maximum power transfer theorem is a fundamental principle in circuit theory stating that maximum power is delivered to the load when the load resistance equals the internal resistance of the source. This condition optimizes the trade-off between voltage drop and current flow, ensuring that power dissipation across the load is maximized.

Optimization and Boundary Conditions

The process of determining the load resistance that maximizes power involves optimization techniques such as setting the derivative of the power with respect to resistance to zero. Additionally, evaluating the behavior of the circuit at extreme values of the load resistance (very low or very high) helps understand why there is a peak value for power. When the load resistance is either too small or too large compared to the internal resistance, the power transferred to the load diminishes, reinforcing the need for matching resistances.

Load Resistor and Circuit Analysis

In an electrical circuit, the load resistor is the external resistance to which power is delivered. Understanding how current flows through the circuit, which includes both the load resistor and the internal resistance of the battery, is crucial for analyzing how power is distributed and ultimately dissipated across the components.

Internal Resistance of a Battery

Every real battery features an internal resistance that affects the overall performance of the circuit. This internal resistance causes a voltage drop within the battery and limits the maximum current that can flow, thereby impacting the power available to the load resistor.

Transcript

00:01 So for this question we have a resistor big r that is attached to a battery with some internal resistance small r.

00:09 We want to find the power dissipated by the load resistor.

00:15 And this is actually goes to, you can calculate as i square r because we are very easily fine why it is i based on the emf and the total resistance of the circuits which is equals to e square over plus r.

00:35 Square times r and we want to find, suppose that this big r can vary what would be the value of r which will give it the maximum power dissipation.

00:51 So to find that we will have to differentiate p with respect to the resistance of the resistor big r and we want to find the maximum point, the point of maximum dispower dissipation, which is when the differentiation is a 0, right? it's equals to 0.

01:14 That's the turning point, which is also the maximum point.

01:19 So differentiating this, you're going to use chain rule.

01:27 So we differentiate the numerator first, and we are left with the denominator.

01:31 Then we differentiate the denominator, and we equate this to 0.

01:52 So simplifying this, we try to make it to the same denominator.

01:57 J have r plus r e square minus 2 e square r and simplifying this further we get e square so this is the term right and for this to be 0 that's only one possibility that is physical that is when r over here is equals to small r and then this term will be 0, so the numerator is 0 and everything turns to 0.

02:58 So therefore, pmax occurs when r is equal to small r, which is basically the internal resistance of the battery.

03:12 Now to put it into a numerical case, we have emf of 9 volts, and we have this is small r to be 1 oms...

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