In measuring a voltage, a voltmeter uses some current from...

In measuring a voltage, a voltmeter uses some current from the circuit. Consequently, the voltage measured is only an approximation to the voltage present when the voltmeter is not connected. Consider a circuit consisting of two $1550-\Omega$ resistors connected in series across a $60.0-\mathrm{V}$ battery. $\quad$ (a) Find the voltage across one of the resistors. (b) A nondigital voltmeter has a full-scale voltage of 60.0 $\mathrm{V}$ and uses a galvanometer with a full-scale deflection of 5.00 $\mathrm{mA}$ . Determine the voltage that this voltmeter registers when it is connected across the resistor used in part (a).

Key Concepts

Internal Resistance of Measuring Instruments

Real-world measuring instruments such as voltmeters have a finite internal resistance, which means that they draw a small amount of current when connected to a circuit. This property affects the circuit by altering the effective resistance and can lead to measurement errors, particularly when the instrument's resistance is not significantly higher than that of the circuit.

Loading Effect

The loading effect refers to the influence a measuring instrument has on the circuit it is measuring due to its finite internal resistance. When a voltmeter is connected across a component, it can draw additional current, thereby changing the voltage distribution in the circuit and causing the measured voltage to deviate from the true value present in the undisturbed circuit.

Voltage Divider

The voltage divider is a fundamental concept in circuit analysis, describing how the total voltage in a series circuit is divided among the components proportionally to their resistances. This concept allows one to calculate the voltage across a particular resistor by considering its resistance relative to the total series resistance.

Ohm's Law

Ohm's law is central to understanding electrical circuits, stating that the voltage across a resistor is equal to the product of its resistance and the current flowing through it (V = IR). This relationship is instrumental in determining how current and voltage distribute in a circuit.

Transcript

00:01 So in this problem, we have two registers of resistances, 1550 -oom, each connected in series across a voltage of 60 -volt battery.

00:12 Now, in part a, we want to find the voltage across each register, or at least one register.

00:22 So first of all, the total resistance in this case is going to be rs equals r plus rr, just like here they are connected in series so the total resistance is going to be r plus r which is going to be equal to 2r all right now the current let's just write it this down as well 30 100 oom now the current is going to be i equals voltage over r s and that's going to be equal to v over 2r so i'll just keep it right now this way and the voltage across one of the registers, let's say we want to measure this register over here.

01:12 Let's say we want to measure this one over here.

01:17 Then let's call this one as vr.

01:21 Then the voltage across that register would be i times r, and the value of i is v over 2r.

01:31 So let's put it there, times r.

01:34 And these two are are cancel out so it's going to be v over 2 which is 60 by 2 volt which is 30 bolt i'm just going to write that so it's going to be 30 volt all right so now in part 2 we're using a galvanometer with full -scale deflection of ig max equals 5 amps 5 malleaps so let's do that first here in part b we have a galvanometer with ig max equals 5m aft all right and it has a full -scale voltage of 60 om when using as a voltmeter okay so v max equals 60 volt and it is being used as a voltage of one of those registers in part a so one of this register over here okay so let's find out what voltage would this voltmeter show remember in part a we found out actually the voltage of each register would be 30 but when we use the voltmeter there's going to be some loss so the voltmeter will not show the actual voltage there will be a little deficiency okay so let's find that out first of all for maximum settings the voltage volt meter resistance would be rv equals b max that's the resistance of the voltmeter rv it's going to be b max over i g max okay and this is going to be 60 volt over 5 times 10 to the negative 3 amps that's the current and this is going to be 1 .2 times 10 to the 4 om all right now this rv here which is connected across the register r is going to be 1 .2 times to the 4 om here.

03:53 Now since these two are in parallel let's first find that equivalent resistance.

03:58 So that equivalent resistance rp is going to be equal to rv times r over rv plus r and we just have the value of everything.

04:19 So i'm just going to plug this in.

04:22 Okay.

04:23 So this is basically 1 .2 times 10 to the 4 om times 1550 om over 1 .2 times 10 to the 4 om plus 1550 oom...

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