A battery with ℰ=6.00 V and no internal resistance supplies...

A battery with $\mathcal{E}=6.00 \mathrm{~V}$ and no internal resistance supplies current to the circuit shown in Figure $\mathrm{P} 27.9 .$ When the double-throw switch $\mathrm{S}$ is open as shown in the figure, the current in the battery is $1.00 \mathrm{~mA}$. When the switch is closed in position $a,$ the current in the battery is $1.20 \mathrm{~mA}$. When the switch is closed in position $b$, the current in the battery is $2.00 \mathrm{~mA}$. Find the resistances (a) $R_{1}$ (b) $R_{2},$ and (c) $R_{3}$.

Key Concepts

Series and Parallel Resistor Networks

Resistors in a circuit can be arranged in series, where the total resistance is the sum of the individual resistances, or in parallel, where the total resistance is found by taking the reciprocal of the sum of the reciprocals of the individual resistances. Understanding these configurations is crucial for determining the equivalent resistance seen by the EMF, which in turn affects the overall current in the circuit.

Circuit Analysis with Switches

Switches are used in circuits to alter the connectivity of components, thereby changing the circuit's overall resistance and current distribution. Analyzing circuits with switches involves determining how the closing or opening of a switch modifies the paths available for current flow, leading to different effective resistor combinations and resulting in different battery currents.

Battery and Electromotive Force

This concept refers to the battery as a device that converts chemical energy into electrical energy, characterized by its electromotive force (EMF) – the voltage it provides when no current is flowing. In circuit analyses, the EMF is a primary source that establishes the potential difference driving the current through the circuit elements.

Ohm's Law

Ohm's Law is the fundamental relationship in electric circuits that relates voltage (V), current (I), and resistance (R) by the equation V = IR. This law is used to determine unknown quantities in a circuit when the other two are known, and it is essential for analyzing how changes in circuit configuration (such as opening or closing a switch) affect the current distribution.

Transcript

00:01 Hi there, so for this problem, we have the circuit that is shown in here.

00:08 We are given the nph associated with this battery that is equal to six poles and no internal resistance.

00:18 Now, when the double -throw switch is open, the current in the battery is one micro -ampers.

00:27 When the cich is closed, at position a, the current is...

00:31 And the current in the battery is 1 .20 micropers and when the switch is closed at position b, the current in the battery is 2 microampers.

00:41 So we need to find the values for the resistance, 1, 2, and 3.

00:51 Okay, so in order to solve this, we are going to consider each case.

00:58 So first of all, when the switch s is opened, we will have the following.

01:10 So we note that then the resistor 1, the resistor 2, and the resistor 3 are in series.

01:23 As you can see from the figure.

01:26 And then we will have the following expression that the sum of the resistance, the resistance 1 plus the resistance 2 plus the resistance 3 is equal to the battery, which is, six balls divided by the current.

01:45 In this case, we are told that that is one micro -ampers.

01:50 So that will be 10 to the minus 3 -ampers.

01:54 So from this, we obtain a value of 6 kilo -mbs.

02:02 Now, for the situation, when s is close in position a, then we will have that.

02:19 We will have a parallel combination of the two resistors two, parallel.

02:29 And this is in series with the resistor 1 and the resistor 3.

02:41 And of course the battery.

02:44 So we will have that this is the resistor, the resistance 1 plus 1 divided by 2 times that resistors 2, which is the equivalent capacitance for that parallel, and this plus the resistance 3.

03:00 Now, this again is 6 poles divided by now the current, we are told that it is 1 .2 times 10 to the minus 3.

03:10 Ampers...

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