Consider a circuit that consists of a real battery with an emf εand an internal resistance of r

Consider a circuit that consists of a real battery with an...

Consider a circuit that consists of a real battery with an emf $\varepsilon$ and an internal resistance of $r$ connected to a variable resistor $R$. (a) In order for the terminal voltage of the battery to be equal to the emf of the battery, what should the resistance of the variable resistor be adjusted to? (b) In order to get the maximum current from the battery, what should the resistance variable resistor be adjusted to? (c) In order for the maximum power output of the battery to be reached, what should the resistance of the variable resistor be set to?

Key Concepts

Current in Series Circuits

In series circuits, the total resistance is the sum of the internal resistance of the battery and the load resistance. Ohm’s law states that the current is equal to the emf divided by this total resistance. Minimizing the load resistance increases the current, but it also leads to different voltage drops and power distributions, which is essential when analyzing conditions for maximum current and voltage outputs.

Maximum Power Transfer Theorem

The maximum power transfer theorem states that maximum power is delivered to a load when the load resistance matches the internal resistance of the source. In the case of a battery connected to a variable resistor, adjusting the resistor to equal the internal resistance ensures that the load receives the maximum possible power, balancing the power lost in the internal resistance and the power delivered to the load.

Real Battery with Internal Resistance

A real battery is modeled as an ideal voltage source (emf) in series with an internal resistance. This internal resistance causes a voltage drop when current flows, meaning that the terminal voltage available to an external circuit is less than the emf. Understanding this model is crucial for analyzing how the battery interacts with a connected load, such as a variable resistor.

Terminal Voltage

Terminal voltage is the voltage measured across the terminals of a battery and is given by the battery's emf minus the voltage drop across its internal resistance. In circuits, the terminal voltage determines the effective voltage that is applied to external components, and achieving a terminal voltage equal to the emf implies that no current is drawn from the battery, thereby eliminating any voltage drop across the internal resistance.

Transcript

00:01 In this question we have a circuit of a battery with some internal resistance, little r, over here, connected to an external load or an external variable resistor, big r.

00:18 We want to find for different conditions what is our r in order to satisfy that condition.

00:25 So for the first condition is that we want the terminal voltage of the battery, that is over here.

00:35 This is the terminals and the terminal voltage to be equals to the emf.

00:44 What does this mean? right, this means that there must be no potential drop, no potential drop across the internal resistance.

01:06 And for that to occur, when there's no potential drop, we know that v equals to i.

01:13 So when there's no potential drop it means that current is zero and for current to be zero we will need r to be infinite so that the overall resistance of the entire circuit is infinite then our current will not flow at all and we will thus have the terminal voltage of our battery to be equals to the emf.

01:48 So basically it's an open circuit.

01:51 It can be an open circuit or r and must be infinite.

01:57 Next to get the maximum current from our battery.

02:06 Again we want to use the equation r equals to v over r.

02:14 For our current to be a maximum.

02:20 Remember that our r effective is just big r plus small r.

02:27 So this will be the entire current of our circuit.

02:33 For current to be increased to the maximum, our r must decrease...