A battery has an emf of 9.20 V and an internal resistance of 1.20 Ω. (a) What resistance across

A battery has an emf of 9.20 V and an internal resistance of...

Key Concepts

Electromotive Force (EMF) and Battery Model

EMF is the open-circuit voltage produced by a battery or any electrical source. In circuit analysis, a battery is modeled as an ideal voltage source in series with an internal resistance. This model helps in understanding how the actual voltage across a load is less than the nominal EMF because of the voltage drop across the internal resistance when current flows.

Internal Resistance

Internal resistance is the inherent resistance within a battery that causes part of the EMF to be lost as heat during current flow. It influences the performance of the battery by limiting the current it can deliver and affects the power available to an external load, making it a crucial factor in analyzing real-world circuits.

Ohm’s Law

Ohm’s Law, which states that the voltage across a resistor is equal to the product of the current flowing through it and its resistance (V = IR), is fundamental in circuit analysis. It provides the basic relationship needed to relate EMF, load resistance, and internal resistance in order to calculate current and voltage drops across circuit elements.

Electrical Power in Resistive Circuits

The power delivered to a resistor in a circuit is given by formulas such as P = I²R or P = V²/R. This concept is essential when determining how much power is extracted from a battery by a load resistor. By considering the voltage drop across the load and the current flowing through it, one can calculate and design for the desired power transfer.

Maximum Power Transfer Theorem

The Maximum Power Transfer Theorem states that maximum power is delivered to a load when the load resistance equals the internal resistance of the source. This concept is critical to understanding the limits of power extraction from a battery. It also provides insight into why certain power outputs may be unattainable if they exceed the maximum possible power transfer based on the battery's internal characteristics.

Transcript

00:01 So what we have here is a circuit where we have a battery with some internal resistance, and we want to come up with an equation that relates the resistance of this circuit element that's attached to the battery and the power that it's extracting.

00:14 So what we can do is we can start off with this equation for power, which is i squared times r.

00:22 So we can rewrite the current i in terms of the emf and these two resistances.

00:28 So this is what that'll look like.

00:31 We'll have epsilon over big r plus little r, since they're in series.

00:39 And we'll square that and multiply it by r.

00:43 So what we can do now is we can put this expression in the form of a quadratic equation.

00:51 And in case if you forgot what the form of a quadratic equation looks like, i'll just go ahead, read the general form on the side here.

01:03 And so if we go ahead, apply that equation to this expression up here.

01:08 For power, what we get is r squared plus 2r minus e squared or epsilon squared, excuse me, divided by p times r minus little r squared.

01:25 And all that equal to zero.

01:27 So now if we want to solve for r, which we want to do in the following parts of this problem, we're going to go ahead and use the quadratic formula, which is a way to solve for x when we're given a quadratic equation.

01:41 And so just to rewrite that quadratic formula for you, it will be minus b plus or minus the square root of b squared, minus 4 times a c, all divided by 2a.

02:03 So now if we go ahead and write the expression where we're solving for big r, we'll have minus 2 .4, minus e squared divided by p, plus or minus square root of 2 .4, minus e squared divided by p, squared minus 5 .76.

02:38 And this is going to be divided by 2.

02:41 So now that we have an expression where we can solve for big r, given these different power points that we're looking for...

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