Question: In Fig. 27-57,R1=2.00R, the ammeter resistance is zero, and the battery is ideal. What multiple ofε/Rgives the current in the ammeter?

Short Answer

Expert verified

Answer:

The multiple of ε/Rthat gives the current in the ammeter is 0.143.

Step by step solution

01

The given data

The given resistance value,R1 = 2R

Ammeter resistance is zero.

02

Understanding the concept of voltage, current and resistance

When there is no voltage drop across the ammeter, then the current across each of the bottom resistors is the same and thus, the total current across the battery is the sum of current across both the resistors. And now calculating the equivalent resistance for the given circuit combination, we can get the current across each bottom resistor and also the given resistance by using Kirchhoff's voltage law. Kirchhoff’s voltage law states that in any closed loop network, the total voltage around the loop is equal to zero. Now, using all the calculated data, we can get the required ammeter reading.

Formulae:

The voltage equation using Ohm’s law, V = I R (1)

Kirchhoff’s voltage law, closed loopV=0 (2)

The equivalent resistance for series combination of the resistors, Req=inRi (3)

The equivalent resistance for parallel combination of the resistors, Req=in1Ri (4)

03

Calculation of the value of ammeter reading

It is noted that there is no voltage drop across the ammeter. Thus, the currents in the bottom resistors are the same that is given by .

So, the current through the battery is given by and the voltage drop across each of the bottom resistors is given using equation (1) as:

The equivalent resistance of the given combination is given using equations (3) and (4) as follows:

Req=2RR2R+R+RRR+R=7R6

Now, the current across the battery can be given using equation (1) as follows:

2i=εReqi=ε27R/6i=3ε7R

Using equation (2), the voltage equation for the left loop can be given as follows:

Loop across the resistorsR1,R

()ε-i2R2R-iR=0i2R=ε-iR2Ri2R=ε-3ε/7RR2Ri2R=4ε/72Ri2R=2ε7R

Thus, the current across the ammeter that is the ammeter reading can be given using equation (1) as follows:

iammeter=i-i2R=3ε7R-2ε7R=ε7R

Hence, the required value of the multiple of ε/Ris 0.143.

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