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In Fig. 27-69,R1=20.0Ω,R2=10.0Ω , and the ideal battery has emf ε=120V. What is the current at point a if we close (a) only switchS1, (b) only switches S1andS2, and (c) all three switches?

Short Answer

Expert verified

a) The current at point a if we close only switchS1is 3.0A.

b) The current at point a if we close only switches S1and S2is 3.75A.

c) The current at point a if we close all the three switches is3.94A .

Step by step solution

01

The given data

a) The value of the resistances given are: R1=20.0Ω R2=10.0Ω

b) Emf of the ideal battery,ε=120V

02

Understanding the concept of current flow

For a closed circuit, the current flowing is the same through the loop. No, in the given case individual switches play their role by contributing the additional resistances upon being close to the equivalent resistance. Thus, using the equivalence equation can find the resistance that can be further used to calculate the required current.

Formula:

The voltage equation using Ohm’s law, V=IR (i)

The equivalent resistance for a series combination,Req=inRn (ii)

The equivalent resistance for a parallel combination, Req=in1Rn (iii)

03

Calculation of the current at point a if only switch S1 is closed

(a)

If we close only switchS1, the resistor 1 only two resistors in series within the closed loop comes into play. Thus, using the given data in equation (i), the current value can be given as:

i=ε2R1=120V220.0Ω=3.0A

Hence, the current value is3.0A.

04

Calculation of the current at point a if only switch S1 and S2 is closed

(b)

If we close only switchesS1 and S2, then the equivalent resistance of the formed circuit can be given using equations (ii) and (iii) as follows:

Req=R1+R1R1+R22R1+R2=20.0Ω+20.0Ω20.0Ω+10.0Ω220.0Ω+10.0Ω=32.0Ω

Now, the current at point a if we close only switchesS1 and S2is given using equation (i) and above equivalent resistance as follows:

i=120.0V32.0Ω=3.75A

Hence, the value of the current is3.75A .

05

Calculation of the current value if all the switches are closed

(c)

If all the three switches are closed, the equivalent resistance of the given resistors can be calculated using equations (ii) and (iii) as follows:

Req=R1+R1R'/R1+R'

where,

R'=R2+R1R1+R22R1+R2

=10.0Ω+20.0Ω20.0Ω+10.0Ω220.0Ω+10.0Ω=22.0Ω

Thus, the equivalent resistance can be given as follows:

Req==20.0Ω+20.0Ω+22.0Ω20.0Ω+22.0Ω=30.5Ω

Now, the current for the condition can be calculate using equation (i) as follows:

i=120V30.5Ω=3.94A

Hence, the value of the current is3.94A.

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Most popular questions from this chapter

Figure shows a circuit of four resistors that are connected to a larger circuit. The graph below the circuit shows the electric potential V(x) as a function of position xalong the lower branch of the circuit, through resistor 4; the potential VAis 12.0 V. The graph above the circuit shows the electric potential V(x) versus position x along the upper branch of the circuit, through resistors 1, 2, and 3; the potential differences areΔVB2.00 V andΔVC5.00 V. Resistor 3 has a resistance of 200 Ω. What is the resistance of (a) Resistor 1 and (b) Resistor 2?

(a) In Fig. 27-18a, with,R1>R2is the potential difference across more than, lessR2than, or equal to that acrossR1?

(b) Is the current through resistor R2 more than, less than, or equal to that through resistorR1?

In Figure,ε=12 V, R1=2000Ω, R2=3000Ω, and R3=4000Ω. (a) What is the potential difference VAVB?(b) What is the potential difference VBVC?(c) What is the potential differenceVCVD?(d) What is the potential difference VAVC?

Figure shows a resistor of resistance R= 6.00 Ω connected to an ideal battery of emf12.0 V by means of two copper wires. Each wire has length 20.0 cm and radius 1.00 mm. In dealing with such circuits in this chapter, we generally neglect the potential differences along the wires and the transfer of energy to thermal energy in them. Check the validity of this neglect for the circuit of Figure: What is the potential difference across (a) The resistor and (b) Each of the two sections of wire? At what rate is energy lost to thermal energy in (c) The resistor And (d) Each section of wire?

In Figure, the ideal batteries have emfsε1=10.0Vandε2=0.500ε1 , and the resistances are each 4.00Ω.

(a) What is the current in resistance 2?

(b) What is the current in resistance 3?

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